RITA CATARINA
CORREIA GUERRA
GENERALIZAC
¸ ˜
OES DA TRANSFORMAC
¸ ˜
AO DE
FOURIER E SUAS APLICAC
¸ ˜
OES
GENERALIZATIONS OF THE FOURIER
TRANSFORM AND THEIR APPLICATIONS
RITA CATARINA
CORREIA GUERRA
GENERALIZAC
¸ ˜
OES DA TRANSFORMAC
¸ ˜
AO DE
FOURIER E SUAS APLICAC
¸ ˜
OES
GENERALIZATIONS OF THE FOURIER
TRANSFORM AND THEIR APPLICATIONS
Tese apresentada `a Universidade de Aveiro para cumprimento dos requisitos necess´arios `a obten¸c˜ao do grau de Doutor em Matem´atica Aplicada, rea-lizada sob a orienta¸c˜ao cient´ıfica do Doutor Lu´ıs Filipe Pinheiro de Castro, Professor Catedr´atico do Departamento de Matem´atica da Universidade de Aveiro, e do Doutor Nguyen Minh Tuan, Professor Associado do Departa-mento de Matem´atica da Escola de Educa¸c˜ao da Universidade Nacional do Vietname
Apoio financeiro da FCT e do FSE no ˆambito do III Quadro Comunit´ario de Apoio, Bolsa de Doutoramento PD/BD/114187/2016.
Professor Catedr´atico da Universidade de Aveiro
Doutora Helena Maria Narciso Mascarenhas
Professora Auxiliar da Universidade de Lisboa
Doutora Maria Teresa Mesquita Cunha Machado Malheiro
Professora Auxiliar da Universidade do Minho
Doutor Alberto Manuel Tavares Sim˜oes
Professor Auxiliar da Universidade da Beira Interior
Doutora Anabela de Sousa e Silva
Investigadora da Universidade de Aveiro
Doutor Lu´ıs Filipe Pinheiro de Castro
compreens˜ao constantes ao longo destes anos.
Ao meu coorientador, Professor Nguyen Minh Tuan, que apesar da distˆancia, conseguiu estar sempre presente, pela supervis˜ao, partilha de conhecimen-tos, apoio e incentivo.
Ao Centro de Investiga¸c˜ao e Desenvolvimento em Matem´atica e Aplica¸c˜oes (CIDMA), pelo acolhimento e facilidades concedidas para a realiza¸c˜ao deste trabalho.
`
A Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) pelo apoio financeiro concedido.
Ao Departamento de Matem´atica da Universidade de Aveiro pelas condi¸c˜oes de trabalho proporcionadas.
Aos colegas com quem me cruzei ao longo deste percurso, pelas ex-periˆencias, alegrias e desabafos partilhados.
Aos meus amigos pelo encorajamento, compreens˜ao, paciˆencia e apoio du-rante estes anos.
`
A minha fam´ılia, por me acompanhar sempre ao longo desta caminhada, pelo apoio incondicional, pela paciˆencia e compreens˜ao das ausˆencias, pela preocupa¸c˜ao e encorajamento constantes.
Ao Diogo, um agradecimento muito especial, por acreditar em mim, por me fazer acreditar que ´e poss´ıvel, por me incentivar a seguir sempre em frente, pela paciˆencia, pela compreens˜ao, pela tolerˆancia, por continuar a caminhar comigo, lado a lado, ao longo destes anos.
invertibilidade, espetro, operador unit´ario, involu¸c˜ao, equa¸c˜ao integral, iden-tidade de Parseval, teorema de Wiener, teoria Tauberiana, princ´ıpio de in-certeza de Heisenberg, filtragem, s´erie de Fourier, fun¸c˜ao de Hermite. MSC 2010 33C45, 40E05, 42A38, 42A85, 43A32, 43A50, 44A20, 44A35, 45A05,
45E10, 46E25, 47A05, 47A10, 47G10, 94A12, 94A20
Resumo Nesta tese, consideramos uma nova generaliza¸c˜ao da transforma¸c˜ao de Fourier, dependente de quatro parˆametros complexos e de todas as potˆencias da transforma¸c˜ao de Fourier. Esta nova transforma¸c˜ao ´e estudada em alguns espa¸cos de Lebesgue. De facto, tendo em conta os valores dos parˆametros, podemos ter n´ucleos muito diferentes e assim, o correspondente operador ´e estudado em diferentes espa¸cos de Lebesgue, de acordo com o seu n´ucleo.
Come¸camos com a caracteriza¸c˜ao de cada operador pelo seu polin´omio caracter´ıstico. Esta caracteriza¸c˜ao serve de base para o estudo das propriedades seguintes. Seguindo isto, apresentamos, para cada caso, o espetro do correspondente operador, condi¸c˜oes necess´arias e suficientes para as quais o operador ´e invert´ıvel, identidades do tipo de Parseval e condi¸c˜oes para as quais o operador ´e unit´ario e uma involu¸c˜ao de ordem n. Depois disto, constru´ımos novas convolu¸c˜oes associadas `aqueles operadores e obtemos as correspondentes identidades de fatoriza¸c˜ao e algumas desigualdades da norma. Usando estes novos operadores e convolu¸c˜oes, constru´ımos novas equa¸c˜oes integrais e estudamos a sua solvabilidade. Neste sentido, temos equa¸c˜oes geradas pelos operadores estudados e tamb´em uma classe de equa¸c˜oes do tipo de convolu¸c˜ao dependendo de fun¸c˜oes de Hermite multidimensionais. Al´em disso, estudamos a solvabilidade de equa¸c˜oes integrais cl´assicas, usando os novos operadores e convolu¸c˜oes, nomeadamente uma classe de equa¸c˜oes de Wiener-Hopf mais Hankel, cuja solu¸c˜ao ´e escrita em termos de uma s´erie do tipo de Fourier. Para um caso desta generaliza¸c˜ao da transforma¸c˜ao de Fourier, que depende apenas das transforma¸c˜oes de Fourier do cosseno e do seno, obtemos resultados de Paley-Wiener e resultados Tauberianos de Wiener, usando a convolu¸c˜ao associada e uma nova transla¸c˜ao induzida por essa convolu¸c˜ao. Princ´ıpios de incerteza de Heisenberg para os casos unidimensional e multidimensional s˜ao obtidos para um caso particular do operador introduzido.
No final, como uma aplica¸c˜ao fora da matem´atica, obtemos um novo resultado em processamento de sinal, mais propriamente, num processo de
property, involution, integral equation, Parseval identity, Wiener theorem, Tauberian theory, Heisenberg uncertainty principle, filtering, Fourier series, Hermite function.
MSC 2010 33C45, 40E05, 42A38, 42A85, 43A32, 43A50, 44A20, 44A35, 45A05, 45E10, 46E25, 47A05, 47A10, 47G10, 94A12, 94A20
Abstract In this thesis, we consider a new generalization of the Fourier transform, depending on four complex parameters and all the powers of the Fourier transform. This new transform is studied in some Lebesgue spaces. In fact, taking into account the values of the parameters of the operator, we can have very different kernels and so, the corresponding operator is studied in different Lebesgue spaces, accordingly with its kernel.
We begin with the characterization of each operator by its charac-teristic polynomial. This characterization serves as a basis for the study of the forthcoming properties. Following this, we present, for each case, the spectrum of the corresponding operator, necessary and sufficient conditions for which the operator is invertible, Parseval-type identities and conditions for which the operator is unitary and an involution of order n. After this, we contruct new convolutions associated with those operators and obtain the corresponding factorization identities and some norm inequalities. By using these new operators and convolutions, we construct new integral equations and study their solvability. In this sense, we have equations generated by the studied operators and also a class of equations of convolution-type depending on multi-dimensional Hermite functions. Furthermore, we study the solvability of classical integral equations, using the new operators and convolutions, namely a class of Wiener-Hopf plus Hankel equations, whose solution is written in terms of a Fourier-type series. For one case of this generalization of the Fourier transform, that only depends on the cosine and sine Fourier transforms, we obtain Paley-Wiener and Paley-Wiener’s Tauberian results, using the associated convolution and a new translation induced by that convolution. Heisenberg uncertainty principles for the one-dimensional case and for the multi-dimensional case are obtained for a particular case of the introduced operator.
At the end, as an application outside of mathematics, we obtain a new result in signal processing, more properly, in a filtering processing, by applying one of our new convolutions.
List of Symbols iii
1 Introduction 1
1.1 Basic background . . . 6
1.1.1 Function spaces . . . 6
1.1.2 Some properties of linear operators . . . 6
1.1.3 The Fourier transform . . . 7
2 Operators generated by Fourier integral operators 9 2.1 Characteristic polynomials . . . 9
2.2 General properties of the operators of the form T(ϑ,ι,κ,$), with ϑ 6= 0 . . . 25
2.2.1 Case T(a,b,c,d) . . . 26 2.2.2 Case T(a,b,c,0) . . . 34 2.2.3 Case T(a,b,b+d,d) . . . 36 2.2.4 Case T(a,0,c,0) . . . 40 2.2.5 Case T(a,0,c,d) . . . 42 2.2.6 Case T(a,b,0,d) . . . 45 2.2.7 Case T(a,0,d,d) . . . 47 2.2.8 Case T(a,d,0,d) . . . 50 2.2.9 Case T(a,c,c,0) . . . 52 2.2.10 Case T(a,0,0,d) . . . 54
2.3 New convolutions associated with the operators of the form T(ϑ,ι,κ,$), with ϑ 6= 0 56 2.4 General properties of the operators of the form T(0,ι,κ,$) . . . 72
2.4.1 Case T(0,b,c,d) . . . 72 2.4.2 Case T(0,b,c,0) . . . 75 2.4.3 Case T(0,b,b+d,d) . . . 77 2.4.4 Case T(0,0,c,0) . . . 80 2.4.5 Case T(0,0,c,d) . . . 80 2.4.6 Case T(0,b,0,d) . . . 82 2.4.7 Case T(0,0,d,d) . . . 82 2.4.8 Case T(0,d,0,d) . . . 84 2.4.9 Case T(0,c,c,0) . . . 85 2.4.10 Case T(0,0,0,d) . . . 85
3.1.1 Equations generated by the operators of the form T(ϑ,ι,κ,$) with two
eigenvalues . . . 96
3.1.2 Equations generated by the operators of the form T(ϑ,ι,κ,$) with three eigenvalues . . . 97
3.1.3 Equations generated by the operators of the form T(ϑ,ι,κ,$) with four eigenvalues . . . 99
3.2 New convolutions associated with integral equations of Wiener-Hopf plus Han-kel type . . . 101
3.2.1 Preliminaries . . . 101
3.2.2 Four new convolutions and norm inequalities . . . 105
3.2.3 Application: Solvability of Wiener-Hopf plus Hankel equations . . . . 109
3.3 Convolution-type integral equations weighted by Hermite functions . . . 112
3.3.1 Auxiliary machinery . . . 112
3.3.2 Even case . . . 115
3.3.2.1 New convolutions for the even case . . . 115
3.3.2.2 Application: solvability of an integral equation . . . 126
3.3.3 Odd case . . . 136
3.3.3.1 New convolutions for the odd case . . . 136
3.3.3.2 Application: solvability of an integral equation . . . 138
4 Paley-Wiener and Wiener’s Tauberian results for oscillatory integral oper-ators 141 4.1 Auxiliary Machinery . . . 142
4.2 Paley–Wiener theorems . . . 143
4.3 Wiener’s Tauberian theorems . . . 148
5 Heisenberg uncertainty principles for an oscillatory integral operator 159 5.1 Uncertainty principles . . . 160
6 Signal processing 167 6.1 Filtering with other transforms . . . 167
6.2 Multiplicative filter related with our transform . . . 169
6.2.1 Examples of application of the designed filter . . . 170
7 Conclusion 175
(τωf )(x) = f (x − ω), ω ∈ Rn, 141
(f )∗n convolution ∗ of n elements f , 135 ¯
z complex conjugate of z ∈ C, 28
cas(±x) = cos(x) ± sin(x), 142
δjk Kronecker delta, 10
`2 Hilbert space of all convergent-square complex series, 103 ˆ
fc(n) Fourier cosine coefficients of f , 102 ˆ
fs(n) Fourier sine coefficients of f , 102 Im{z} imaginary part of z ∈ C, 48 h·, ·i2 usual inner product in L2, 8
[λ1; λ2; λ3; λ4; λ5; λ6] representation of the operator T(ϑ,ι,κ,$) in terms of projectors, in the sense that each entry of [λ1; λ2; λ3; λ4; λ5; λ6] is the coefficient corre-sponding to each projector P1, P2, P3, Q1, Q2 and Q3, respectively, 14
C set of complex numbers, 6
N set of natural numbers, 5
N0 = N ∪ {0}, 11
Q quadratic-phase Fourier transform, 3
R set of real numbers, 2
A = L1(Rn), equipped with the convolution multiplication ∗ T(0,b,c,0)
, 142 D(Rn) space of smooth functions with compact support, 6
Ef Fourier-type series of f on [0, 2π] corresponding to the kernel E, 103
T (f) linear hull of all translationsTωf , with ω ∈ Rn, 151
Tωf translation of f by ω ∈ Rn related with the convolution ∗ T(0,b,c,0)
, 150 W = T(0,b,c,0)(L1(Rn)) equipped with the usual L1-norm, 151
W0 = T(0,b,c,0)(X0) ⊂W , 151
X = L1(Rn), equipped with each one of the convolution multiplications •
(j)and ?(j), with j = 1, 2, 3, 4, 125 E kernel of the series Ef, 102
Hk Hermite polynomial of degree k, 11 S Cauchy singular integral transform, 9 Pj
k=0Mj,k sum of all terms of the form Mj,k, such that j and k have the same parity, 31
Φα multi-dimensional Hermite function of order kαk, 11 φk Hermite function of degree k, 11
ψk normalized Hermite function of degree k, 11
Re{z} real part of z ∈ C, 28
σ(K) spectrum of the operator K, 6
spanT(f ) closed linear hull of translationsTωf of f , 151 supp(f ) support of the function f , 147
Pj
k=0Mj,k sum of all terms of the form Mj,k, such that j and k have different parities, 31
~ (j)
, j = 1, 2, 3, 4 convolutions weighted by multi-dimensional Hermite functions asso-ciated to the operator T(0,b,c,0)−1 , 137
(j)
, j = 1, 2, 3, 4 convolutions weighted by multi-dimensional Hermite functions asso-ciated to the operator T(0,b,c,0), 136
?
(j), j = 1, 2, 3, 4 convolutions weighted by multi-dimensional Hermite functions asso-ciated to the operator T(0,b,c,0)−1 , 120
•
(j), j = 1, 2, 3, 4 convolutions weighted by multi-dimensional Hermite functions asso-ciated to the operator T(0,b,c,0), 115
F ⊗
Tj,Tk,T`
, j, k, ` = 1, 2 two parameter finite Fourier-type convolutions associated with the operators T1 and T2, 105
∗ T(ϑ,ι,κ,$)
convolution associated with the operator T(ϑ,ι,κ,$) in the sense that (T(ϑ,ι,κ,$)(f ∗
T(ϑ,ι,κ,$)
g))(x) = (T(ϑ,ι,κ,$)f )(x)(T(ϑ,ι,κ,$)g)(x), for all x ∈ Rn, this is, ∗
T(ϑ,ι,κ,$)
satisfies the T(ϑ,ι,κ,$)–factorization, 56 Ψ = ιTc+ κTs+ $Tc2, with ι, κ, $ ∈ C, 26
kf kp usual Lp-norm of f , 1 ≤ p ≤ ∞, 6 e
ϕ(x) = ϕ(−x), x ∈ Rn, 88
B(x0, ) open ball centered at x0 ∈ Rn with radius , 142 C∞(Rn) space of all infinitely differentiable functions on Rn, 6
C0(Rn) space of all continuous functions that tend to zero at infinity, 6 C0m(Rn) space of all m-differentiable complex-valued functions on Rnthat
van-ish at infinity, 6
Dxαf derivative of order |α| of f , 11
F Fourier transform, 7
Ff Fourier series of f , 102
Fn Fourier coefficients of f , 104 F(a,b,c,d) linear canonical transform, 2 Fθ fractional Fourier transform, 2
H1 Hartley transform with kernel cos(·) + sin(·), 9 H2 Hartley transform with kernel cos(·) − sin(·), 9
I identity operator, 6
J Hankel transform, 9
K∗ adjoint operator of the operator K, 7 K−1 inverse operator of the operator K, 6
L(X) set of all linear operators defined from X to X, 9
Lp(Rn) Lebesgue space of pth-power integrable functions on Rn(1 ≤ p < ∞), 6
Pj, j = 1, 2, 3 projectors associated with the operator Tc, 10 PK characteristic polynomial of the operator K, 9 Qj, j = 1, 2, 3 projectors associated with the operator Ts, 10 Q(a,b,c) quadratic Fourier transform, 2
R(j)(a,b,c,d), j = 1, 2, 3, 4 projectors associated with the operator T(a,b,c,d), 30
rin input signal, 168
rout output signal, 168
T = aI + bTc+ cTs+ dTc2, with a, b, c, d ∈ C, 12 T0 = a0I + b0F + c0F2+ d0F3, with a0, b0, c0, d0 ∈ C, 12
T1 Finite two parameters Fourier-type transform defined with kernel 1
b cos(nx) + 1
csin(nx), n ∈ Z, 103
T2 finite two parameters Fourier-type transform defined by (T2f )(n) := (T1f )(−n), for every n ∈ Z, 104
Tc cosine Fourier transform, 10
Ts sine Fourier transform, 10
T(0,0,0,d) = dTc2, with d 6= 0, 85 T(0,0,c,0) = cTs, with c 6= 0, 80 T(0,0,c,d) = cTs+ dTc2, with cd 6= 0, c 6= −d and c 6= d, 80 T(0,0,d,d) = dTs+ dTc2, with d 6= 0, 82 T(0,ι,κ,$) = ιTc+ κTs+ $Tc2, with ι, κ, $ ∈ C, 72 T(0,b,0,d) = bTc+ dTc2, with b 6= 0, b 6= −d and b 6= d, 82 T(0,b,b+d,d) = bTc+ (b + d)Ts+ dTc2, with bcd 6= 0 and b 6= −d, 77 T(0,b,c,0) = bTc+ cTs, with bc 6= 0, b 6= −c and b 6= c, 75 T(0,b,c,d) = bTc+ cTs+ dTc2, with bcd 6= 0, c 6= −(b + d), c 6= b + d, c 6= −(b − d) and c 6= b − d, 72 T(0,c,c,0) = cTc+ cTs, with c 6= 0, 85 T(0,d,0,d) = dTc+ dTc2, with d 6= 0, 84 T(ϑ,ι,κ,$) = ϑI + ιTc+ κTs+ $Tc2, with ϑ, ι, κ, $ ∈ C, 12
T(a,0,c,0) = aI + cTs, with ac 6= 0, 40
T(a,0,c,d) = aI + cTs+ dTc2 with acd 6= 0, c 6= −d and c 6= d, 42 T(a,0,d,d) = aI + dTs+ dTc2, with ad 6= 0, 47
T(a,ι,κ,$) = aI + ιTc+ κTs+ $Tc2, with a ∈ C\{0}, ι, κ, $ ∈ C, 26 T(a,b,0,d) = aI + bTc+ dTc2, with ab 6= 0, b 6= −d and b 6= d, 45 T(a,b,b+d,d) = aI + bTc+ (b + d)Ts+ dTc2, with abd 6= 0 and b 6= −d, 36 T(a,b,c,0) = aI + bTc+ cTs, with abc 6= 0, b 6= −c and b 6= c, 34
T(a,b,c,d) = aI + bTc+ cTs + dTc2, with abcd 6= 0, c 6= b + d, c 6= −(b + d), c 6= b − d and c 6= −(b − d), 26
T(a,c,c,0) = aI + cTc+ cTs, with ac 6= 0, 52 T(a,d,0,d) = aI + dTc+ dTc2, with ad 6= 0, 50
W reflection operator, this is, (W ϕ)(x) :=ϕ(x) = ϕ(−x), x ∈ Re n, 88 X0 subspace of L1(Rn) such that T(0,b,c,0)f has compact support, 151 FrFT fractional Fourier transform, 1
Introduction
The Fourier transform appeared by the hand of Joseph Fourier. According to some au-thors, his greatest contribution was to show that a function f defined on a finite interval can be written in terms of a trigonometric series. The work of Fourier where we can find this result is Th´eorie Analytique de la Chaleur, 1822. However, this work was been presented to the French Academy of Science in 1807, but it was not accepted, because of the lack of rigor of its proofs. Although we attribute the idea of the Fourier transform to Joseph Fourier, actually, that idea was known by other mathematicians, as Cauchy, Laplace or Poisson (cf. [39, 117]). Fourier transform has been studied over the years by many authors. In fact, we can see the Fourier transform in different contexts, namely in mathematics and physics. Most of the authors report the importance of the Fourier transform in these two areas. If, on the one hand, we can see the Fourier transform being used associated with many problems in physics, on the other hand, e.g., Vretblad [109] states that its beauty attracts the mathematicians. In [9], Bracewell points out some areas in which the Fourier transform is used, namely in electrical engineering, biology, medicine and remote sensing. This author emphasize the fact that, in physics, the Fourier transform be seen as a relationship between concepts. Besides this, some abstract results, from the mathematical point of view, related with the Fourier transform properties correspond to simple phenomena in physics.
Throughout of the years, the class of integral transforms of Fourier-type has been studied for more than a century. There is a very extensive list of their applications and operators generated by them (see [61, 62, 104, 113]). Nevertheless, this type of transforms continues to be a topic of interest for many researchers, namely in what concerns to generalizations and applications (cf. [1, 22, 55, 61, 76]). This is the case of the fractional Fourier transform (FrFT), that appears in the twenties and thirties of the last century, from quantum mechanics (see [62, 112, 113]).
Following this, we can point out the works of N. Wiener in 1929, H. Weyl in 1930, E.U. Condon in 1937, H. Kober in 1939 [62], A. P. Guinand in 1956, A. L. Patterson in 1959, V. Bargmann in 1961, De Bruijn in 1973 and R. S. Khare in 1974. Anyway, in the first part of the last century, there were not so many researchers interested in these generalizations, until Namias [72]. Namias, in eighties, reinvented the FrFT, still in the context of quantum mechanics. His work was formalized by A. C. McBride and F. H. Kerr in 1987 [68] and the fractional operators were modified with a view to avoid some ambiguities. A detailed state of the art of the generalizations of the Fourier transform, in terms of considering the corresponding powers with fractional exponents, can be find in [82]. Briefly, we can say that
the FrFT is a one parameter transform that has the Fourier transform as a particular case. This transform is defined as follows
(Fθf )(x) = Z
R
kθ(x, y)f (y) dy, (1.1)
where kθ(x, y) = q√3(θ) 2πe
iq1(θ)(x2+y2−2q2(θ)xy), if θ is not a multiple of π δ(y − x), if θ is a multiple of 2π δ(y + x), if θ + π is a multiple of 2π, with q1(θ) = cot(θ) 2 , q2(θ) = sec(θ), q3(θ) = √ i − cot θ and δ is the Dirac delta.
The research related with this transform has grown in the last years, being constructed new convolutions with some purposes in what concerns to applications, namely in signal processing (see, e.g., [2, 5, 78, 79, 80, 81, 93]).
Besides the FrFT, we would like to emphasize the linear canonical transform (LCT). The LCT is a four real parameters linear integral transform, being three of those parameters free and the other one is a constraint and we can define it as follows
(F(a,b,c,d)f )(x) = q −i 2πbe i2bdx2R Re −i1 bxyei a 2by2f (y) dy, if b 6= 0 √ deicd2x 2 f (dx), if b = 0, (1.2) where a b c d
is a matrix with determinant equal to 1. This class of integral transforms arose from paraxial optics and quantum mechanics, separately, but simultaneously, during the seventies of the last century (see [70, 114]). This family of transforms includes the FrFT, being that the growing interest in the FrFT, during the 1990s, has revived the interest in LCT, from other points of view. The LCT generalizes several transforms in the sense that the Fourier transform, the Fractional Fourier transform, the Fresnel transform, scaling operations and multiplication by chirp functions can be seen as particular cases of LCT. During the last years, many works related with new convolutions associated with the LCT have been published and the biggest interest in study these convolutions is to apply, for example, to signal processing (cf. [37, 50, 51, 110, 111]). In fact, the LCT has many applications in signal processing, design of filters, signal separation, pattern recognition and so on (see, e.g., [7, 82, 92]). In [7], it was proved that the optimal filtering with LCT is better than with the FrFT, in the sense that the mean-square error (MSE) in recovering a signal distorted by a noise is less for the process with the LCT.
More recently, in 2014, Castro et al. [22] studied six cases of another generalization of the Fourier transform, that is the quadratic Fourier transform. This transform is defined by
(Q(a,b,c)f )(x) = Z
R
e−ix(ay2+by+c)f (y) dy, (1.3) where a, b, c ∈ R. As we can see, this transform depends on three real parameters and was obtained by considering a simple heat equation and applying the theory of reproducing
kernels. In fact, the idea came from the fact of generalize the exponent of the transform associated with the mentioned solution of the heat equation.
Later, in 2018, L. P. Castro, L. T. Minh and N. M. Tuan (the first and the third are two of the authors of [22]) introduced an even more general transform, called quadratic-phase Fourier transform, that include as subcases the FrFT and LCT (cf. [24]). This transform is defined by
(Qf )(x) = √1 2π
Z R
eiq(a,b,c,d,e)(x,y)f (y) dy, (1.4) where
q(a,b,c,d,e)(x, y) = ax2+ bxy + cy2+ dx + ey,
with a, b, c, d, e ∈ R and b 6= 0. The authors derived some basic properties of this new transform and constructed new convolutions associated with it. Some Young-type inequalities and norm decay rate of oscillatory integrals were obtained as well as necessary and sufficient conditions for the unique solvability of new integral equations, generated by the constructed convolutions.
So, we can see that this type of integral transforms allows us to obtain convolutions and derive integral equations. New convolutions are very useful integral transforms which exhibit as many applications as many properties they will allow. Although this is a classical subarea of mathematical analysis, it continues to flourishing precisely because of its high potential and concrete applications. Convolutions and integral equations of convolution-type occupy a central position both in mathematical theoretical studies and in applications to other sciences. Indeed, on the one hand, convolutions are effectively applied in many practical problems (even outside mathematics), on the other hand, each convolution is itself a new integral transform and so, subjected to a mathematical theoretical analysis (cf., e.g., [6, 8, 16, 24, 25, 28, 29]). For instance, the Hilbert and Cauchy integral transforms can be interpreted as convolutions in some sense. This is why despite decades of research, convolutions and their associated equations continue to be vigorously developed by many authors, and time-to-time become powerful tools in engineering applications (see [4, 36, 98] and references therein).
Regarding the convolutions, one of the main topics related with them is getting norm inequalities. This type of inequalities has been studied along of the years by some researchers, as E. M. Stein, that presented three inequalities for the Fourier convolution, just considering the spaces L1(Rn) and L∞(Rn). R. O’neil, considering the work of Stein and the Lorentz spaces, proved a norm inequality for functions in Lorentz spaces (cf. [77]). In some other studies, there is the application of other methods with weighted Lebesgue spaces in which is possible to obtain norm inequalities among some Lp-spaces. This is the case of the works [30, 38, 74], in which, at least in one of them, are also using methods of reproducing kernel Hilbert spaces. The present thesis proposes a different approach although containing some of the goals of those other works.
The convolutions can also be seen as a tool that we can apply inside or outside mathe-matics. Inside of mathematics, a very strong application of the convolutions is related with their factorization identities. In fact, applying the corresponding operator, we can decouple the convolutions into a product of two integrals, weighted or not by some other function. Moreover, we also have integral equations of convolution-type, that are generated somehow by convolutions, that, as we said before, can be seen as integral transforms. Such equations arise often from one of two possible ways: the first is from the modeling situations in terms of differential operators/equations, and the second one is from the so-called convolution
in-tegral problems (see [3, 35, 58, 102] and references therein). One of the main reasons for the existence of such studies is that the solutions of those types of equations have many useful applications in diverse fields such as: scattering theory, fluid dynamics, filtering theory, cir-cular punch penetration in finitely thick elastic layers, rarefied gas dynamics, linear filtering of stationary random processes and in the field of fast signal processing.
Related with the above mentioned operators, we can also study Paley-Wiener and Wiener’s Tauberian results. In fact, those results are important structural results in harmonic analysis. Basically, we can know the asymptotic behavior of some functions, through other objects whose properties are better known. On the other hand, Tauberian results can be the converse of other results, that by themselves can not be true, but, when we add an extra condition, called by Rudin [91] a tauberian condition, they become true. Along of the years, many works have been published in this sense (see, e.g., [63, 64, 65]).
Another topic of relevant interest is related with the Heisenberg uncertainty principles. The uncertainty principle of the Fourier transform states that a nonzero function and its Fourier transform can not both be essentially localized. This result is applied in quantum mechanics, signal processing and information theory. The idea of these uncertainty principles comes from Heisenberg, being improved and generalized by other authors (see, e.g., [41, 42, 44, 53, 57, 60, 67, 90, 95, 96]). These uncertainty principles are important in the interpretation of the behavior of integral operators and in the theory of partial differential equations.
The main aim of this thesis is to introduce a new generalization of the Fourier trans-form, depending on four complex parameters and all the powers of Fourier transform (that is equivalent to say that depends on the identity operator and on the cosine and sine Fourier transforms). According to the value of each parameter, we can have very different kernels and the idea is to study some basic properties of these different cases and then apply the corresponding operators. Moreover, we introduce new convolutions associated with those operators that we will also apply after. In terms of applications, we have applications in mathematics, such as the solvability of integral equations, the obtaining of Paley-Wiener and Wiener’s Tauberian results for a class of operators and Heisenberg uncertainty principles. Outside of mathematics, we present an application of a particular operator to signal process-ing, more precisely, we construct a filter, based on the introduced convolution associated with that operator.
In comparison with the other transforms mentioned above, fractional Fourier transform, linear canonical transform, quadratic Fourier transform and quadratic-phase Fourier trans-form, in which the generalization is due to a change of the exponent of the exponential function that appears in the kernel of the Fourier transform, in our case, we generalize the Fourier transform by considering a linear combination of four operators that coincide with all the powers of the Fourier transform. So, actually, as the reader can see in the next chapter, we will have a part of the operator corresponding to a linear combination of the identity operator and the reflection operator and an integral part, whose kernel is a linear combination between the kernel of the Fourier transform and the kernel of its inverse.
This thesis is divided into seven chapters and organized as follows.
In Chapter 1, that is the present one, we have some basic background, that will be helpful to understand the concepts and results introduced in the following chapters. We begin by introducing some function spaces that we will be working on in this thesis. After this, we present some properties of linear operators and, more specifically, some properties of the Fourier transform. We begin by presenting the definition of the Fourier transform in different spaces (S (Rn), L1(Rn) and L2(Rn)), the inversion theorem in these different spaces and
some other properties, as the spectrum of the operator, the Parseval identity, the convolution theorem and the Young inequality.
In Chapter 2, we start by identifying the operator that we study in this thesis, which cor-responds to a generalization of the Fourier transform. This operator is generated univocally by the Fourier transform and its powers. However, we rewrite it in a different way (depending univocally on the identity operator and the cosine and sine Fourier transforms). This is done to simplify some computations. This operator depends on four complex parameters and, de-pending on these parameters, we have different properties for the operator. We start with the determination of the characteristic polynomial associated with each case of the parameters. After this, using these characteristic polynomials, we are able to study some properties asso-ciated with each case of the operator. For the cases in which the parameter assoasso-ciated with the identity operator is different from zero, we study the corresponding operators in L2(Rn). For the other cases, we study the corresponding operators in L1(Rn) and L2(Rn). For each case, we present the definition of the operator in the spaces in which we define it and its polynomial identity, the spectrum of the operator and the inversion theorem(s), if applicable. Parseval-type identities, necessary and sufficient conditions for which the operator is unitary and necessary and sufficient conditions for which the operator is an involution of order n ∈ N are also obtained. For all the cases for which the operator is invertible, we construct convo-lutions and the corresponding factorizations identities, in L2(Rn). For one of the cases, that only depends on the cosine and sine Fourier transforms, we obtain a Young-type inequality.
Chapter 3 is devoted to integral equations. In this chapter, we consider three different types of integral equations. The first class of equations considered is generated by an operator and we can solve it taking profit of the projectors, in the sense of the Lagrange interpolation formula, associated with the operator that generates the equation. The second class of equa-tions considered is of Wiener-Hopf plus Hankel type, defined in a finite interval. To obtain sufficient conditions for which this type of equations has a unique solution, we construct a set of four convolutions that we will apply to the solvability of those equations. The third class of equations is of convolution-type and depends on the multi-dimensional Hermite functions. The idea to solve this type of equations is similar to the previous one, but in this case, we construct convolutions weighted by multi-dimensional Hermite functions.
In Chapter 4, we obtain Paley–Wiener and Wiener’s Tauberian results associated with an oscillatory integral operator, which depends on cosine and sine kernels. Additionally, a new Wiener-type algebra is also associated with the new convolution related with that operator (introduced in Chapter 1). At the end of the chapter, a new Wiener-Pitt-type Theorem is deduced.
Chapter 5 gives us Heisenberg uncertainty principles for a specific oscillatory integral operator which representatively exhibits different parameters on their sine and cosine phase components. We begin by considering these principles for the one-dimensional case and then, we generalize them for the multi-dimensional case.
In Chapter 6, an application of that type of operators outside of mathematics is considered. The idea is to obtain a new result in filtering of signals. Some existing examples are analyzed and then we present two examples of application of a particular operator to signal processing, more specifically, we construct a low-pass filter to recover a signal distorted by some noise.
This thesis ends with a brief conclusion, in which we make a balance of the most repre-sentative results that we achieve.
The results in this thesis are mainly based on the author’s papers [15, 16, 17, 18, 19, 20, 21] and those of other authors are properly referred.
1.1
Basic background
In this section, we provide some tools that are used throughout the thesis. In particular, we introduce some well-known definitions and results that can be found in some classical books of functional analysis and operator theory. In this thesis, we are studying generalizations of the Fourier transform, that is a very important tool in functional analysis and its applications. Thereby, we will recall the definition of that operator and some properties in some different spaces. To begin, we will introduce the spaces that we will use in this thesis and some properties of linear operators that we will study in this thesis.
1.1.1 Function spaces
We denote by C∞(Rn) the space of all infinitely differentiable functions on Rn.
S (Rn) is denoting the Schwartz space, that is the space of all infinitely differentiable functions on Rn, whose derivatives tend to zero at infinity faster than any inverse power of x. For 1 ≤ p < ∞, Lp(Rn) defines the Banach space of all Lebesgue measurable complex valued functions defined on Rn for which
Z Rn
|f (x)|pdx < ∞. (1.5)
We endow this space with the norm kf kp := R
Rn|f (x)| pdx1p
and we call it norm of f ∈ Lp(Rn).
If p = ∞, we have the Banach space L∞(Rn), endowed with the norm kf k∞:= ess supx∈Rn|f (x)|,
that is the space of all essentially bounded and Lebesgue measurable complex valued functions defined on Rn.
C0(Rn) is being used to denote the space of all continuous functions that tend to zero at infinity, that is a subspace of L∞(Rn).
Let D(Rn) be the space of all infinitely differentiable functions with compact support and C0m(Rn) the space of all m-differentiable complex-valued functions on Rnthat vanish at infinity.
1.1.2 Some properties of linear operators
In this subsection, we will consider
K : X → X
a linear operator defined from X to X, where X is a Banach space.
Now, we will present some properties of this type of operators, that will be useful in the following chapters. In the sequel, I is denoting the identity operator and K−1 denotes the inverse operator of K.
Definition 1.1.1 (Spectrum of an operator). The spectrum of a linear operator K, σ(K), is the set of values λ ∈ C such that (K − λI)−1 does not exist.
Definition 1.1.3 (Unitary operator). A bounded linear operator K is said to be unitary if it is bijective and K∗ = K−1, where K∗ is the adjoint operator of K.
In this work, we will be using the notation K0= I.
Definition 1.1.4 (Involution of order n). The operator K is an involution of order n ∈ N if Kn= I and Kj 6= I, for every j = 1, . . . , n − 1. If n = 2, we say that K is an involution.
1.1.3 The Fourier transform
The Fourier transform, F , can be defined by the following integral, since it is well defined, (F f )(x) = 1
(2π)n2 Z
Rn
e−ixyf (y) dy, (1.6)
where xy denotes the usual inner product of x and y, with x, y ∈ Rn. This transform is studied in many books of functional analysis (see, e. g. [39, 52, 117], that we used to do the compilation of some properties of the Fourier transform) and can be defined in different, but equivalent, ways. In this thesis, we will consider the just presented form (1.6).
This transform is often presented as an operation in L1(Rn). However, when we define it in L1(Rn), we have some problems when we want to recover the function f from F f . In fact, if f ∈ L1(Rn), we have the following result.
Lemma 1.1.5 (Riemann-Lebesgue Lemma). F is a continuous linear operator of L1(Rn) into C0(Rn) and fulfills the following norm inequality
kF f k∞≤ 1 (2π)n2
kf k1. (1.7)
This result does not impliy that F f is an integrable function. So, to define the inverse Fourier transform, we need to add an extra condition (we need to impose that F f ∈ L1(Rn)), as we can see in the following result.
Theorem 1.1.6 (Inversion Theorem of F in L1(Rn)). Let f ∈ L1(Rn) and F f ∈ L1(Rn), then 1 (2π)n2 Z Rn eixy(F f )(y) dy = f (x), (1.8)
for almost every x ∈ Rn.
However, if we define the Fourier transform in the Schwartz space,S (Rn), we have that, in that space, in addition of being one-to-one, the Fourier transform is surjective and so, we can define the inverse Fourier transform of any function inS (Rn).
Theorem 1.1.7 (Inversion Theorem of F inS (Rn)). If f ∈S (Rn), then 1 (2π)n2 Z Rn eixy(F f )(y) dy = f (x), (1.9) for every x ∈ Rn.
Having the Fourier transform defined in S (Rn), we can extend its definition to L1(Rn) and L2(Rn). However, we have the mentioned problems when defining the inverse of F in L1(Rn). Concerning to L2(Rn), the integral that defines the Fourier transform does not converge absolutely for functions in L2(Rn). However, we can define the Fourier transform in L1(Rn) ∩ L2(Rn), being an isometry with respect to the L2-norm. As L1(Rn) ∩ L2(Rn) is a dense subspace of L2(Rn), we can define a unique extension of the Fourier transform on L2(Rn), that we will also denote by F for simplicity. As S (Rn) is dense in L2(Rn), we can conclude that F is also bijective in L2(Rn). So, we can define the inverse operator of F in L2(Rn) as follows
Theorem 1.1.8 (Inversion Theorem of F in L2(Rn)). If f ∈ L2(Rn), then 1
(2π)n2 Z
Rn
eixy(F f )(y) dy = f (x), (1.10) for almost every x ∈ Rn.
In L2(Rn), we can state the following properties.
Proposition 1.1.9. The spectrum of the operator F is given by σ(F ) = {±1, ±i}.
Theorem 1.1.10 (Parseval identity). If f, g ∈ L2(Rn), then hF f, F gi2 = hf, gi2, where h·, ·i2 represents the usual inner product in L2(Rn).
Corollary 1.1.11. If f ∈ L2(Rn), then kF f k2= kf k2.
Theorem 1.1.12. Considering the Fourier transform, F , and (f ∗ Fg)(x) = 1 (2π)n2 Z Rn f (τ )g(x − τ ) dτ, (1.11)
where f and g are defined over Rn, we have the following factorization identity (F (f ∗
Fg))(x) = (F f )(x)(F g)(x), (1.12)
for all x ∈ Rn.
Theorem 1.1.13 (Young inequality). If f ∈ Lp(Rn), g ∈ Lq(Rn) and 1 p + 1 q = 1 r + 1, with 1 ≤ p, q, r ≤ ∞, then f ∗ F g ∈ L r(Rn) and kf ∗ Fgkr≤ kf kpkgkq. (1.13)
Operators generated by Fourier
integral operators
Due to the importance of the Fourier transform in mathematical analysis and outside of mathematics, throughout of the years, several operators of Fourier-type have been studied. The most known are the Laplace transform, the Hartley transforms, the Hilbert transform and so on. But, more recently, new Fourier-type transforms have appeared as generalizations of the Fourier transform. In this chapter, we propose a new generalization of the Fourier transform and study some of its basic properties. Moreover, we construct new convolutions associated with that.
2.1
Characteristic polynomials
It is well-known that several of the most important integral transforms are involutions when considered in appropriate spaces. For instance, the Hankel transform J , the Cauchy singular integral operator S on a closed curve and the Hartley transforms (typically denoted by H1 and H2, see [9, 10, 11, 47, 105]) are involutions of order 2. Moreover, the Fourier transform F and the Hilbert transform H are involutions of order 4 (i.e. H4 = I, simply becauseH is an anti-involution in the sense that H2= −I).
Those involution operators possess several significant properties that are useful for solving problems which are somehow characterized by those operators, as well as several kinds of integral equations, and ordinary and partial differential equations with transformed argument (see [8, 23, 28, 31, 32, 34, 42, 48, 49, 54, 67, 69, 72, 84, 85, 99, 101, 109]).
In order to have a global view on corresponding linear operators, we start by recalling the concept algebraic operators.
Let L(X) be the space of all linear operators defined from X to X.
Definition 2.1.1. An operator K ∈ L(X) is said to be algebraic if there exists a (non-zero) polynomial PK(t) = tN + µ1tN −1+ . . . + µN −1t + µN, with µj ∈ C, j = 1, . . . , N, such that PK(K) = 0. Moreover, the algebraic operator K is said to be of order N if PK(K) = 0 for a polynomial PK, and Q(K) 6= 0, for any polynomial Q of degree less than N . In such a case, PK is said to be the characteristic polynomial of K (and its roots are called the characteristic roots of K).
As an example, we may directly refer to the characteristic polynomial of some of the above mentioned integral operators as follows:
PJ(t) = t2− 1; PS(t) = t2− 1; PH1(t) = PH2(t) = t
2− 1; P
H(t) = t2+ 1 (see [6, 66, 69, 87, 88, 89, 108]).
Due to their importance in the applications, this section is devoted to the determination of characteristic polynomials of some integral operators which are generated by the Fourier integral operator F . In particular, the Fourier integral operator that we are considering in here is defined in the Hilbert space L2(Rn) as in (1.6). As above mentioned, it is well-known that the operator F is an involution of order 4 (thus F4 = I, where I is the identity operator on L2(Rn)). In other words, F is an algebraic operator which has a characteristic polynomial given by PF(t) = t4−1. Such polynomial has obviously the following four characteristic roots: 1, −i, −1, i.
In addition, we can write F = Tc− iTs, where (Tcf )(x) := 1 (2π)n2 Z Rn cos(xy)f (y)dy and (Tsf )(x) := 1 (2π)n2 Z Rn sin(xy)f (y)dy are called the cosine Fourier and sine Fourier transforms, respectively.
We know that Tc3 = Tc and Ts3 = Ts, so PTc(t) = PTs(t) = t
3− t, whose characteristic roots are −1, 0, 1. We also know that Tc2+ Ts2= I and TcTs= TsTc= 0. Therefore, based on that information, we are able to consider three projectors corresponding to Tc
P1= I − Tc2, P2= T2 c − Tc 2 , P3= T2 c + Tc 2 , (2.1)
which satisfy the following identities PjPk= δjkPk, for j, k = 1, 2, 3, P1+ P2+ P3 = I, Tc= −P2+ P3, where δjk is the Kronecker delta.
We also have three projectors corresponding to Ts Q1= I − Ts2, Q2= Ts2− Ts 2 , Q3= Ts2+ Ts 2 , (2.2)
which satisfy the following identities QjQk= δjkQk, for j, k = 1, 2, 3, Q1+ Q2+ Q3= I, Ts= −Q2+ Q3. Moreover, we have F = − P2+ P3+ iQ2− iQ3, (2.3) F2 =P2+ P3− Q2− Q3, (2.4) F3 = − P2+ P3− iQ2+ iQ3, (2.5) F4 =P2+ P3+ Q2+ Q3 = I. (2.6)
It is also clear that
β1P1+ γ1P2+ δ1P3+ β2Q1+ γ2Q2+ δ2Q3 = 0 if and only if
β1 = γ1= δ1 = β2 = γ2= δ2= 0.
Those six projectors will be very useful in what follows. Namely, when we will be consid-ering our classes of algebraic operators generated by Tcand Ts.
Let us also recall the classical Hermite functions. For the one-dimensional case (see e.g. [100, 101]), the Hermite polynomial of degree k can be defined by
Hk(x) = (−1)kex 2 d dx k e−x2, k ∈ N0, (2.7) and we write φk(x) := e− 1 2x 2 Hk(x) = (−1)ke 1 2x 2 d dx k
e−x2, that is a solution of the following differential equation
d2y dx2 − x
2y = −(2k + 1)y, (2.8)
which is equivalent to the Schr¨odinger equation for a harmonic oscillator in quantum me-chanics, as these functions are their eigenfunctions. For the corresponding orthogonal system on L2(R), we take ψk(x) := φk(x)/(2kk!
√
π)12. The Hermite functions form an orthonormal basis of L2(R) and they are closely related to the Whittaker function.
More generally, the multi-dimensional Hermite functions are defined by Φα(x) := (−1)|α|e
1 2|x|
2
Dxαe−|x|2, x ∈ Rn,
where α = (α1, . . . , αn) is an n-tuple of non-negative integers αk, k = 1, . . . , n, |α| := α1+· · ·+ αnand Dαxf := ∂
αf
∂x1α1···∂xnαn represents the derivative of order |α| of the function f . Theoretically, Hermite functions can be seen as essential particular components of functional analysis by which many objects have been developed (see [100] and references therein). In practice, the Gaussian functions are widely known for describing the normal distributions in statistics, defining Gaussian filters in image and signal processing, solving heat and diffusion equations, generating the Weierstrass transform, etc., while the Hermite functions are related to the
parabolic cylinder and essential functions in harmonic analysis. All of those are concerned to the theory of integral equations.
Namely, we have (see [103, 104])
(TcΦα)(x) = Φα(x), if |α| ≡ 0 (mod 4) 0, if |α| ≡ 1, 3 (mod 4) −Φα(x), if |α| ≡ 2 (mod 4) (2.9) and (TsΦα)(x) = 0, if |α| ≡ 0, 2 (mod 4) Φα(x), if |α| ≡ 1 (mod 4) −Φα(x), if |α| ≡ 3 (mod 4). (2.10)
The next theorem presents the characteristic polynomials of algebraic integral operators generated by the Fourier transform.
In fact, our main goal is to study the operator T0 = a0I + b0F + c0F2 + d0F3, with a0, b0, c0, d0 ∈ C, that is the most general operator generated by the Fourier transform and its powers. However, we will rewrite this operator as follows
T = aI + bTc+ cTs+ dTc2, (2.11) with a, b, c, d ∈ C.
Remark 2.1.2. The operators T and T0are equivalent. If we rewrite the operator F in terms of Tcand Ts, we can obtain the operator T0 written in the following form
T0 =a0(Tc2+ Ts2) + b0(Tc− iTs) + c0(Tc− iTs)2+ d0(Tc− iTs)3 =a0(Tc2+ Ts2) + b0(Tc− iTs) + c0(Tc2− Ts2) + d0(Tc3+ iTs3) =a0(Tc2+ Ts2) + b0(Tc− iTs) + c0(Tc2− Ts2) + d 0(T c+ iTs) =(b0+ d0)Tc+ i(−b0+ d0)Ts+ (a0+ c0)Tc2+ (a0− c0)Ts2 =(b0+ d0)Tc+ i(−b0+ d0)Ts+ (a0+ c0)Tc2+ (a 0− c0 )(I − Tc2) =(a0− c0)I + (b0+ d0)Tc+ i(−b0+ d0)Ts+ 2c0Tc2.
Considering a = a0− c0, b = b0+ d0, c = i(−b0+ d0) and d = 2c0, we obtain the operator T and we can state the following theorem, in which we will use the operator T . This theorem is related with the identification of the characteristic polynomial associated with our operator for different cases of the parameters a, b, c, d ∈ C. In fact, this theorem covers all the possible cases of the parameters for which we have different characteristic polynomials. Further ahead, we will use another notation that will allow us to identify immediately which is the case of the theorem that we are dealing with. We will use T(ϑ,ι,κ,$) to denote the operator T(ϑ,ι,κ,$) = ϑI + ιTc+ κTs+ $Tc2, with ϑ, ι, κ, $ ∈ C, this is, the parameter in the first place denotes the coefficient of the identity operator, I; the parameter in the second place represents the coefficient of the Fourier cosine operator, Tc; the parameter in the third position coincides with the coefficient of the Fourier sine operator, Ts; and the parameter in the fourth position refers to the coefficient of the square of the Fourier cosine operator, Tc2. These different notations are being used for simplicity.
Theorem 2.1.3. Let us consider the operator
T = aI + bTc+ cTs+ dTc2, a, b, c, d ∈ C, (2.12) T is an algebraic operator, whose characteristic polynomial is:
1. a) PT(t) = t2− (2a + d)t + a(a + d) (2.13) if and only if b = c = 0 and d 6= 0; (2.14) b) PT(t) = t2− 2at + a2− c2 (2.15) if and only if d = 0, b = ±c and b 6= 0; (2.16) c) PT(t) = t2− 2(a + d)t + a(a + 2d) (2.17) if and only if c = 0, b = ±d and b 6= 0; (2.18) d) PT(t) = t2− 2at + a2− d2 (2.19) if and only if b = 0, c = ±d and c 6= 0; (2.20) 2. a)
PT(t) = t3− (3a + 2d)t2+(a + d)(3a + d) − b2 t − a2(a + 2d) + a(b2− d2) (2.21) if and only if
c = 0, b 6= −d, b 6= d and b 6= 0; (2.22) b)
PT(t) = t3− (3a + d)t2+ 3a2− c2+ 2ad t − (a2− c2)(a + d) (2.23) if and only if b = 0, c 6= −d, c 6= d and cd 6= 0; (2.24) c) PT(t) = t3− 3at2+ 3a2− c2 t + a(c2− a2) (2.25) if and only if b = d = 0 and c 6= 0; (2.26) d)
PT(t) =t3+ (−3a + b − d) t2+3a2− 2a(b − d) − (b + d)2 t (2.27) +−a3+ (a2− (b + d)2)(b − d) + a(b + d)2 (2.28)
if and only if
bcd 6= 0 and c = ±(b + d); (2.29)
e)
PT(t) =t3− (3a + b + d) t2+3a2+ 2a(b + d) − (b − d)2 t +−a3+ ((b − d)2− a2)(b + d) + a(b − d)2 (2.30) if and only if bcd 6= 0 and c = ±(b − d); (2.31) 3. a) PT(t) =t4− 4at3+ (6a2− b2− c2)t2− 2 2a3− ab2− ac2 t + (a2− c2)(a2− b2) (2.32) if and only if d = 0, b 6= −c, b 6= c and bc 6= 0; (2.33) b)
PT(t) =t4− 2(2a + d)t3+ (6a2− b2− c2+ 6ad + d2)t2− 2 2a3− ab2− ac2 +3a2d − c2d + ad2 t + (a2− c2)(a + d)2− b2
(2.34) if and only if bcd 6= 0 and
c 6= −(b + d) c 6= b + d c 6= −(b − d) c 6= b − d. (2.35)
Proof. Having in mind the form of the projectors introduced above (in (2.1) and (2.2)), we are able to rewrite T in the form
T =aI + bTc+ cTs+ dTc2
T =a Tc2+ Ts2 + bTc+ cTs+ dTc2
=a(P2+ P3+ Q2+ Q3) + b(−P2+ P3) + c(−Q2+ Q3) + d(P2+ P3) = (0) P1+ (a − b + d) P2+ (a + b + d) P3+ (0) Q1+ (a − c) Q2+ (a + c) Q3
= [0; a − b + d; a + b + d; 0; a − c; a + c] , (2.36) where each entry of the last line of (2.36) is the coefficient corresponding to each projector P1, P2, P3, Q1, Q2 and Q3, respectively.
To determine the characteristic polynomial for each case of the operator T , we may begin by considering a polynomial of order 1, that is the characteristic polynomial of the operator T when b = c = d = 0, but in this case we obtain the trivial operator T = aI. In this way, we may consider a polynomial of order 2, that is, PT(t) = t2+ mt + n. This polynomial is the characteristic polynomial of the operator T if and only if PT(T ) = 0 and if there is not a polynomial Q with deg(Q) < 2 such that Q(T ) = 0.
Moreover, the condition PT(T ) = 0 is equivalent to (a − b + d)2+ m(a − b + d) + n = 0 (a + b + d)2+ m(a + b + d) + n = 0 (a − c)2+ m(a − c) + n = 0 (a + c)2+ m(a + c) + n = 0.
The solutions of this system of equations are b = c = d = 0 (which is the trivial operator T = aI), or b = c = 0 d 6= 0 m = −(2a + d) n = a(a + d), which is the case 1. a), or
d = 0 b = ±c b 6= 0 m = −2a n = a2− c2, which is the case 1. b), or
c = 0 b = ±d b 6= 0 m = −2(a + d) n = a(a + 2d), which corresponds to the case 1. c), or
b = 0 c = ±d c 6= 0 m = −2a n = a2− d2, which corresponds to the case 1. d).
Let us prove each case individually. For each case, we need to prove that PT(T ) = 0 and that there does not exist any polynomial Q with degree less than 2 such that Q(T ) = 0. Conversely, we will prove that, for each case, if we have the characteristic polynomial PT, then we obtain the corresponding conditions for the parameters a, b, c, d.
the operator T written in the above form (2.36), we can prove that PT(T ) = 0: T2− (2a + d)T + a(a + d)I =h0; (a + d)2; (a + d)2; 0; a2; a2
i − (2a + d) [0; a + d; a + d; 0; a; a] + a(a + d) ][0; 1; 1; 0; 1; 1]
=[0; 0; 0; 0; 0; 0].
Now, we will prove that there does not exist any polynomial Q with deg(Q) < 2 such that Q(T ) = 0.
Suppose that exists a polynomial Q, defined by Q(t) = t + m such that Q(T ) = 0, this is, (
(a + d) + m = 0 a + m = 0.
This implies that d = 0, which is not under the conditions (2.14). In this way, we can say that PT(t) = t2− (2a + d)t + a(a + d) is the characteristic polynomial of the operator T , when (2.14) holds.
Conversely, assume that (2.13) is the characteristic polynomial of T . Thus, PT(T ) = 0, which is equivalent to
0 =T2− (2a + d)T + a(a + d)I
=0; (a − b + d)2; (a + b + d)2; 0; (a − c)2; (a + c)2
− (2a + d) [0; a − b + d; a + b + d; 0; a − c; a + c] + a(a + d) [0; 1; 1; 0; 1; 1] .
The solutions of this system of equations are b = c = d = 0 (which is the trivial operator T = aI) or b = c = 0. So, case 1. a) is proved.
Case 1. b): If d = 0, b = ±c and b 6= 0, then PT(t) = t2 − 2at + a2− c2. Indeed, by using the operator T written in the above form (2.36) and considering the case b = c (the case b = −c is analogous), it is possible to verify that PT(T ) = 0, as follows
T2− 2aT + (a2− c2)I =0; (a − c)2; (a + c)2; 0; (a − c)2; (a + c)2 − 2a [0; a − c; a + c; 0; a − c; a + c] + a2− c2 [0; 1; 1; 0; 1; 1]
=[0; 0; 0; 0; 0; 0].
Now, we will prove that there does not exist any polynomial Q with deg(Q) < 2 such that Q(T ) = 0.
Suppose that there exists a polynomial Q, defined by Q(t) = t+m, that satisfies Q(T ) = 0. In this case, we would have the following system of equations
(
(a + c) + m = 0 (a − c) + m = 0,
Conversely, assume that PT(t) = t2− 2at + a2− c2 is the characteristic polynomial of T . Thus, PT(T ) = 0, which is equivalent to
0 = T2− 2aT + (a2− c2)I =0; (a − b + d)2; (a + b + d)2; 0; (a − c)2; (a + c)2 − 2a [0; a − b + d; a + b + d; 0; a − c; a + c] + a2− c2 [0; 1; 1; 0; 1; 1] .
The solutions of this system are b = 0 and c = ±d (which corresponds to the case 1. d)) or d = 0 and b = ±c. So, case 1. b) is proved.
Case 1. c): Suppose that c = 0, b = d and b 6= 0 (the case c = 0, b = −d and b 6= 0 is proved in a similar way). We will prove that PT(t) = t2 − 2(a + d)t + a(a + 2d) is the characteristic polynomial of T . Indeed, we can prove that PT(T ) = 0, as follows
T2− 2(a + d)T + a(a + 2d)I =0; a2; (a + 2d)2; 0; a2; a2 − 2(a + d) [0; a; a + 2d; 0; a; a] + a(a + 2d)[0; 1; 1; 0; 1; 1] =[0; 0; 0; 0; 0; 0].
Now, we will prove that there does not exist any polynomial Q with deg(Q) < 2 such that Q(T ) = 0. For that, we will consider that there exists a polynomial Q under these conditions, this is, there exists a polynomial of the form Q(t) = t + m such that Q(T ) = 0. This is equivalent to
(
(a + d) + m = 0 a + m = 0,
which implies that d = 0. However, this is not under the conditions for this operator – (2.18). So, we can conclude that the polynomial (2.17) is the characteristic polynomial of this operator.
Conversely, suppose that the polynomial (2.17) is the characteristic polynomial of T . So, PT(T ) = 0, which is equivalent to
0 =T2− 2(a + d)T + a(a + 2d)I =0; (a − b + d)2; (a + b + d)2; 0; (a − c)2; (a + c)2 − 2(a + d) [0; a − b + d; a + b + d; 0; a − c; a + c]
+ a(a + 2d)[0; 1; 1; 0; 1; 1].
This system of equations is equivalent to b = c = d = 0 (which is the trivial case) or c = 0 and b = ±d. So, case 1. c) is proved.
Case 1. d): Suppose that b = 0, c = d and c 6= 0 (the case b = 0, c = −d and c 6= 0 is proved in a similar way). We will prove that PT(t) = t2− 2at + a2− d2 is the characteristic polynomial of T . Indeed, we can prove that PT(T ) = 0, as follows
T2− 2aT + (a2− d2)I =0; (a + d)2; (a + d)2; 0; (a − d)2; (a + d)2 − 2a [0; a + d; a + d; 0; a − d; a + d] + (a2− d2)[0; 1; 1; 0; 1; 1]
Now, we will prove that there does not exist any polynomial Q with deg(Q) < 2 such that Q(T ) = 0. Suppose that there exists a polynomial of the form Q(t) = t + m such that Q(T ) = 0. This is equivalent to
(
(a + d) + m = 0 (a − d) + m = 0,
which implies that d = 0. But this is not under the conditions of the operator – (2.20). So, we can conclude that the polynomial (2.19) is the characteristic polynomial of this operator. Conversely, suppose that the polynomial (2.19) is the characteristic polynomial of T . So, PT(T ) = 0, which is equivalent to
0 =T2− 2aT + (a2− d2)I
=0; (a − b + d)2; (a + b + d)2; 0; (a − c)2; (a + c)2 − 2a [0; a − b + d; a + b + d; 0; a − c; a + c] + (a2− d2)[0; 1; 1; 0; 1; 1].
This is equivalent to b = c = d = 0 (which is the trivial case) or b = 0 and c = ±d. So case 1. d) is proved.
Now, we will consider a polynomial of degree 3 of the form PT(t) = t3 + mt2+ nt + p. This polynomial is the characteristic polynomial of the operator T if and only if PT(T ) = 0 and if there does not exist any polynomial Q with deg(Q) < 3 such that Q(T ) = 0. That PT(T ) = 0 is equivalent to (a − b + d)3+ m(a − b + d)2+ n(a − b + d) + p = 0 (a + b + d)3+ m(a + b + d)2+ n(a + b + d) + p = 0 (a − c)3+ m(a − c)2+ n(a − c) + p = 0 (a + c)3+ m(a + c)2+ n(a + c) + p = 0.
The solutions of this system of equations are b = c = d = 0 (which is the trivial case T = aI), the solutions corresponding to the cases 1. a), 1. b), 1. c) e 1. d) (which characteristic polynomials are of degree 2), or
c = 0 b 6= −d b 6= d b 6= 0 m = −(3a + 2d) n = (a + d)(3a + d) − b2 p = −a2(a + 2d) + a(b2− d2),
which is the case 2. a), or b = 0 c 6= −d c 6= d cd 6= 0 m = −(3a + d) n = 3a2− c2+ 2ad p = −(a2− c2)(a + d), which is the case 2. b), or
b = d = 0 c 6= 0 m = −3a n = 3a2− c2 p = a(c2− a2), which is the case 2. c), or
bcd 6= 0 c = ±(b + d) m = −3a + b − d n = 3a2− 2a(b − d) − (b + d)2 p = −a3+ (a2− (b + d)2)(b − d) + a(b + d)2, which is the case 2. d), or
bcd 6= 0 c = ±(b − d) m = −(3a + b + d) n = 3a2+ 2a(b + d) − (b − d)2 p = −a3+ ((b − d)2− a2)(b + d) + a(b − d)2, which is the case 2. e).
Similarly to what we have done for the polynomial of order 2, we will prove each case separately.
Case 2. a): If c = 0, b 6= −d, b 6= d and b 6= 0, then
PT(t) = t3− (3a + 2d)t2+(a + d)(3a + d) − b2 t − a2(a + 2d) + a(b2− d2)
we obtain
T3− (3a + 2d)T2+(a + d)(3a + d) − b2 T + −a2(a + 2d) + a(b2− d2) I =0; (a − b + d)3; (a + b + d)3; 0; a3; a3 − (3a + 2d)0; (a − b + d)2; (a + b + d)2; 0; a2; a2 +(a + d)(3a + d) − b2 [0; a − b + d; a + b + d; 0; a; a] +−a2(a + 2d) + a(b2− d2) [0; 1; 1; 0; 1; 1] =[0; 0; 0; 0; 0; 0], that is PT(T ) = 0.
To prove that PT(t) is the characteristic polynomial of T , we have to prove that there does not exist any polynomial Q such that deg(Q) < 3 and Q(T ) = 0. For that, let us consider that there is a such polynomial, this is, exists Q(t) = t2+ mt + n such that Q(T ) = 0. This is equivalent to (a − b + d)2+ m(a − b + d) + n = 0 (a + b + d)2+ m(a + b + d) + n = 0 a2+ ma + n = 0,
which implies that b = d = 0 (which is the trivial operator T = aI) or b = 0 and d 6= 0 (which corresponds to the case 1. a)) or b 6= 0 and b = ±d (which corresponds to the case 1. c)), but any of this solutions is not under the conditions that we have imposed – (2.22).
Conversely, assume that (2.21) is the characteristic polynomial of T . This implies that PT(T ) = 0, which is equivalent to
0 =T3− (3a + 2d)T2+(a + d)(3a + d) − b2 T + −a2(a + 2d) + a(b2− d2) I =0; (a − b + d)3; (a + b + d)3; 0; (a − c)3; (a + c)3
− (3a + 2d)0; (a − b + d)2; (a + b + d)2; 0; (a − c)2; (a + c)2 +(a + d)(3a + d) − b2 [0; a − b + d; a + b + d; 0; a − c; a + c] +−a2(a + 2d) + a(b2− d2) [0; 1; 1; 0; 1; 1].
This is equivalent to c = 0 and b = ±d (which is the case 1. c)) or d = 0 and b = ±c (which is the case 1. b)) or c = 0. So, case 2. a) is proved.
Case 2. b) If b = 0, c 6= −d, c 6= d and cd 6= 0, then
PT(t) = t3− (3a + d)t2+ 3a2− c2+ 2ad t − (a2− c2)(a + d)
is the characteristic polynomial of T . Indeed, using the operator T written in the form (2.36), we can prove that PT(T ) = 0 as follows
T3− (3a + d)T2+ 3a2− c2+ 2ad T − (a2− c2)(a + d)I =0; (a + d)3; (a + d)3; 0; (a − c)3; (a + c)3
− (3a + d)0; (a + d)2; (a + d)2; 0; (a − c)2; (a + c)2 + 3a2− c2+ 2ad [0; a + d; a + d; 0; a − c; a + c] − (a2− c2)(a + d)[0; 1; 1; 0; 1; 1]
To prove that PT(t) is the characteristic polynomial of T , we should prove that there does not exist any polynomial Q such that deg(Q) < 3 and Q(T ) = 0. For that, let us consider that there is a such polynomial, this is, exists Q(t) = t2+ mt + n such that Q(T ) = 0. This is equivalent to (a + d)2+ m(a + d) + n = 0 (a − c)2+ m(a − c) + n = 0 (a + c)2+ m(a + c) + n = 0.
This implies that c = d = 0 (which is the trivial operator T = aI) or c = 0 and d 6= 0 (which is the case 1. a)) or c = ±d and c 6= 0 (which is the case 1. d)). Any solution is not under the conditions (2.24).
Conversely, assume that (2.23) is the characteristic polynomial of T . This implies that PT(T ) = 0, which is equivalent to
0 = T3− (3a + d)T2+ 3a2− c2+ 2ad T − (a2− c2)(a + d)I =0; (a − b + d)3; (a + b + d)3; 0; (a − c)3; (a + c)3
− (3a + d)0; (a − b + d)2; (a + b + d)2; 0; (a − c)2; (a + c)2 + 3a2− c2+ 2ad [0; a − b + d; a + b + d; 0; a − c; a + c] − (a2− c2)(a + d)[0; 1; 1; 0; 1; 1].
This is equivalent to b = 0 and c = ±d (which is the case 1. d)) or d = 0 and b = ±c (which is the case 1. b) or b = 0. So, case 2. b) is proved.
Case 2. c): Suppose that b = d = 0 and c 6= 0. We can prove that, under these conditions PT(T ) = 0, as follows T3− 3aT2+ 3a2− c2 T + a(c2− a2)I =0; a3; a3; 0; (a − c)3; (a + c)3 − 3a0; a2; a2; 0; (a − c)2; (a + c)2 3a2− c2 [0; a; a; 0; a − c; a + c] a(c2− a2)[0; 1; 1; 0; 1; 1] =[0; 0; 0; 0; 0; 0].
To prove that the polynomial (2.25) is the characteristic polynomial of T , we need to prove that there is not any polynomial Q, with deg(Q) < 3, such that Q(T ) = 0. For that, we will consider that such polynomial exists, this is, there exists Q(t) = t2+ mt + n such that Q(T ) = 0, which is equivalent to a2+ ma + n = 0 (a − c)2+ m(a − c) + n = 0 (a + c)2+ m(a + c) + n = 0.
This implies that c = 0 (which is the trivial operator T = aI and is not under the conditions (2.26)). So,
PT(t) = t3− 3at2+ 3a2− c2 t + a(c2− a2) is the characteristic polynomial of T for this case.
Conversely assume that (2.25) is the characteristic polynomial of T . Thus, PT(T ) = 0, this is, 0 =T3− 3aT2+ 3a2− c2 T + a(c2− a2)I =0; (a − b + d)3; (a + b + d)3; 0; (a − c)3; (a + c)3 − 3a0; (a − b + d)2; (a + b + d)2; 0; (a − c)2; (a + c)2 + 3a2− c2 [0; a − b + d; a + b + d; 0; a − c; a + c] + a(c2− a2)[0; 1; 1; 0; 1; 1],
which is equivalent to b = 0 and c = ±d (which is the case 1. d)) or d = 0 and b = ±c (which is the case 1. b)) or d = b and c = ±2b (which is a particular case of the case 2. d)) or d = −b and c = ±2b (which is a particular case of the case 2. e)) or b = d = 0. So, case 2. c) is proved. Case 2. d): Suppose that bcd 6= 0 and c = b + d (the case c = −(b + d) is proved in the same way). We will prove that PT(T ) = 0, using the operator T written in the form (2.36),
T3+ (−3a + b − d) T2+3a2− 2a(b − d) − (b + d)2 T +−a3+ (a2− c2)(b − d) + a(b + d)2 I =0; (a − b + d)3; (a + b + d)3; 0; (a + b + d)3; (a − b − d)3 + (−3a + b − d)0; (a − b + d)2; (a + b + d)2; 0; (a + b + d)2; (a − b − d)2 +3a2− 2a(b − d) − (b + d)2 [0; a − b + d; a + b + d; 0; a + b + d; a − b − d] +−a3+ (a2− c2)(b − d) + a(b + d)2 [0; 1; 1; 0; 1; 1] =[0; 0; 0; 0; 0; 0].
Now, we will prove that there does not exist any polynomial Q, with deg(Q) < 3, such that Q(T ) = 0. For that, we will consider that such polynomial exists and this is equivalent to (a − b + d)2+ (a − b + d)m + n = 0 (a + b + d)2+ (a + b + d)m + n = 0 (a − b − d)2+ (a − b − d)m + n = 0.
This implies that b = d = 0 (which is the case 2. c)) or b = 0 and d 6= 0 (which is the case 1. d)) or d = 0 and b 6= 0 (which is the case 1. b)) or bd 6= 0 and b = −d (which is one of the possibilities of the case 1. c)). These solutions are not under the conditions (2.29). So, the polynomial (2.27) is the characteristic polynomial of T in this case.
Conversely, suppose that (2.27) is the characteristic polynomial of T . Thus, PT(T ) = 0, this is,
0 =T3+ (−3a + b − d) T2+3a2− 2a(b − d) − (b + d)2 T +−a3+ (a2− c2)(b − d) + a(b + d)2 I
=0; (a − b + d)3; (a + b + d)3; 0; (a + b + d)3; (a − b − d)3
+ (−3a + b − d)0; (a − b + d)2; (a + b + d)2; 0; (a + b + d)2; (a − b − d)2 +3a2− 2a(b − d) − (b + d)2 [0; a − b + d; a + b + d; 0; a + b + d; a − b − d] +−a3+ (a2− c2)(b − d) + a(b + d)2 [0; 1; 1; 0; 1; 1],
which is equivalent to c = 0 and b = d (which corresponds to one of the possibilities of the case 1. c)) or c = ±(b + d). So, the case 2. d) is proved.
Case 2. e): The proof of this case can be obtained in a very similar way to the previous one. So, we will omit it.
Now, we will consider a polynomial of order 4 of the form PT(t) = t4+ mt3+ nt2+ pt + q. This polynomial is the characteristic polynomial of the operator T if and only if PT(T ) = 0 and if there does not exist any polynomial Q with deg(Q) < 4 such that Q(T ) = 0. That PT(T ) = 0 is equivalent to
(a − b + d)4+ m(a − b + d)3+ n(a − b + d)2+ p(a − b + d) + q = 0 (a + b + d)4+ m(a + b + d)3+ n(a + b + d)2+ p(a + b + d) + q = 0 (a − c)4+ m(a − c)3+ n(a − c)2+ p(a − c) + q = 0
(a + c)4+ m(a + c)3+ n(a + c)2+ p(a + c) + q = 0.
The solutions of this system of equations are the solutions corresponding to the trivial case (T = aI), to the conditions of the cases 1. a), 1. b), 1. c), 1. d), 2. a), 2. b), 2. c), 2. d), 2. e) (already studied and whose characteristic polynomials are of order less than 4), or
d = 0 b 6= ±c bc 6= 0 m = −4a n = 6a2− b2− c2 p = −2(2a3− ab2− ac2) q = (a2− c2)2(a2− b2), which is the case 3. a), or
bcd 6= 0 c 6= −(b + d) c 6= b + d c 6= −(b − d) c 6= b − d m = −2(2a + d) n = 6a2− b2− c2+ 6ad + d2 p = −2(2a3− ab2− ac2+ 3a2d − c2d + ad2) q = (a2− c2)(a + d)2− b2 ,
which corresponds to the case 3. b).
Similarly to what we have done for the polynomials of orders 2 and 3, we will prove each case separately.
