O estudo dos t´opicos que iremos considerar em seguida est´a desigualmente desenvol- vido. Enquanto que podemos dar como praticamente encerrado o estudo de alguns desses assuntos, o de outros est´a apenas principiado, raz˜ao pela qual n˜ao poderemos ser completamente precisos.
O primeiro t´opico que citaremos ´e o estudo da convergˆencia do algoritmo de inter- pola¸c˜ao e extrapola¸c˜ao em Cn baseado na forma discreta do algoritmo de Papoulis-
Gerchberg. Encontra-se praticamente conclu´ıdo Ferreira (1994b), e inclui a apre- senta¸c˜ao de majorantes e minorantes ´optimos para o erro em fun¸c˜ao do n´umero de itera¸c˜oes, a determina¸c˜ao do valor ´optimo da constante de relaxa¸c˜ao, uma avalia¸c˜ao do efeito da posi¸c˜ao das amostras desconhecidas na velocidade de convergˆencia, e a determina¸c˜ao, para cada tipo de funcional B, das posi¸c˜oes mais favor´aveis e menos favor´aveis para as amostras desconhecidas.
Apesar do algoritmo discreto de Papoulis-Gerchberg ser um m´etodo de dimens˜ao
n, ditada pela dimens˜ao dos vectores de amostras considerados, as conclus˜oes deste
estudo aplicam-se tamb´em ao m´etodo iterativo de dimens˜ao reduzida que estud´amos no terceiro cap´ıtulo do presente trabalho, obtido da equa¸c˜ao u = Su + h pelo m´etodo das aproxima¸c˜oes sucessivas.
Temos tamb´em iniciado o estudo de certos algoritmos um pouco mais complexos mas tamb´em mais poderosos para a solu¸c˜ao iterativa do problema de interpola¸c˜ao e extrapola¸c˜ao, que nalguns casos aceleram substancialmente a convergˆencia Ferreira (1993). Esses algoritmos obtˆem-se com base no particionamento da matriz de itera¸c˜ao dos algoritmos iterativos discutidos no terceiro cap´ıtulo.
Para dar uma melhor ideia das possibilidades destes m´etodos apresentamos um exemplo, obtido a partir do sinal representado na figura 6.1. Trata-se de um vector
6.2. TRABALHO FUTURO 127 0 0.3 0.6 -0.3 -0.6 1 256
Figura 6.1: Vector de dimens˜ao 256 gerado aleatoriamente e limitado em frequˆencia de modo a ter uma largura de banda normalizada igual a 0.5.
0 1 2
-1
1 256
Figura 6.2: Vector amostrador de densidade normalizada 0.7. O processo de amos- tragem correspondente introduz uma perda de amostras de 30 por cento.
1 10 1 10-1 10-2 10-3 10-4 0 10 20 30 40 50 (a) (b) (c) (d) (e)
Figura 6.3: Erro quadr´atico versus n´umero de itera¸c˜oes para os m´etodos (a) Papoulis- Gerchberg, (b) Jacobi, (c) Gauss-Seidel, (d) Jacobi com particionamento, (e) Gauss- Seidel com particionamento.
de dimens˜ao 256, gerado aleatoriamente, e limitado em frequˆencia de modo a ter uma largura de banda normalizada igual a 0.5. O vector amostrador, tamb´em de dimens˜ao 256, est´a representado na figura 6.2. As amostras nulas deste vector definem as posi¸c˜oes das amostras desconhecidas do vector original. A densidade normalizada do vector amostrador ´e 0.7, o que corresponde a um problema de reconstru¸c˜ao em que 30 por cento de amostras do sinal original s˜ao desconhecidas. O n´umero m´aximo de zeros consecutivos ´e 5.
A evolu¸c˜ao do erro para cada um dos m´etodos estudados nos cap´ıtulos anteriores consta da figura 6.3, na qual representamos tamb´em o erro obtido com algoritmos conseguidos por particionamento da matriz de itera¸c˜ao. Como se pode ver, o de- sempenho destes ´ultimos ´e claramente superior. A ordem da matriz de itera¸c˜ao dos m´etodos que vimos no terceiro cap´ıtulo ´e ditada pelo n´umero de inc´ognitas ou amos- tras desconhecidas, que ´e igual a 77 neste caso. O particionamento efectuado foi feito tomando uma matriz de itera¸c˜ao 7 × 7 com blocos 11 × 11.
Com base nos resultados que apresent´amos anteriormente ´e f´acil formular algo- ritmos potencialmente capazes de tirar partido de m´aquinas com capacidade de mul- tiprocessamento. Apesar das boas perspectivas pr´aticas que os resultados de que j´a dispomos parecem sugerir, a sua discuss˜ao foi exclu´ıda do presente trabalho. Espera- mos poder complet´a-la no futuro.
Existe uma outra linha de estudo que n˜ao explor´amos, apesar de tal ser perfeita- mente poss´ıvel. Referimo-nos a problemas de reconstru¸c˜ao multidimensional an´alogos aos que consider´amos. Parece-nos que a via mais curta para a obten¸c˜ao de resul- tados nessa direc¸c˜ao passa pela generaliza¸c˜ao do teorema de Kramer para fun¸c˜oes
f : IRn → IR, a qual ´e, ali´as, trivial. Definindo apropriadamente o conceito de
sobre-amostragem, poder-se-ia ent˜ao chegar a solu¸c˜oes iterativas e n˜ao-iterativas para problemas de reconstru¸c˜ao multidimensional sem grande esfor¸co te´orico. Por brevi- dade, opt´amos por omitir a discuss˜ao deste t´opico, que esperamos poder vir a efectuar numa pr´oxima oportunidade.
N˜ao queremos deixar de referir uma outra possibilidade de trabalho, apesar de n˜ao dispormos ainda de resultados definitivos sobre o assunto. Referimo-nos `a an´alise da robustez dos m´etodos iterativo e n˜ao-iterativo, em presen¸ca de dados inconsisten- tes ou de ru´ıdo. Nestas condi¸c˜oes, como ´e de esperar, o m´etodo n˜ao-iterativo pode conduzir a resultados menos satisfat´orios do que os m´etodos iterativos, os quais s˜ao relativamente f´aceis de controlar por varia¸c˜ao do n´umero de itera¸c˜oes. No entanto, e como o m´etodo n˜ao-iterativo depende de uma opera¸c˜ao de invers˜ao matricial, o seu desempenho depende em boa medida da robustez do m´etodo de invers˜ao utilizado. Nestas condi¸c˜oes os m´etodos baseados em decomposi¸c˜ao em valores singulares pare- cem ser os mais recomend´aveis. Existem, contudo, outras possibilidades, como sejam a utiliza¸c˜ao de pseudo-inversas, que correspondem `a solu¸c˜ao de norma m´ınima no sentido dos m´ınimos quadrados, e a utiliza¸c˜ao de t´ecnicas de regulariza¸c˜ao.
Referˆencias bibliogr´aficas
Louis Auslander e R. Tolimieri. Is computing with the finite Fourier transform pure or applied mathematics? Bull. Am. Math. Soc., 1(6):847–897, 1979. 1.6
S. Banach. Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales. Fundam. Math., 3:133–181, 1922. 3.10, 4.5.1, 4.7
H. Bateman. On the inversion of a definite integral. Proc. Lond. Math. Soc., II. Ser., 4:461–498, 1906. 4.7
R. P. Boas. Entire Functions. Academic Press, New York, 1954. 1.3, 5.5.6, 5.5.6, 5.6 C. B. Boyer. A History of Mathematics. Princeton University Press, Princeton, 1985.
2.4
J. L. Brown, Jr. An RKHS analysis of sampling theorems for harmonic-limited signals.
IEEE Trans. Acoust., Speech, Signal Processing, 33(2):437–440, 1985. 2.3.3
P. L. Butzer. A survey of the Whittaker-Shannon sampling theorem and some of its extensions. J. Math. Res. Expos., 3(1):185–212, 1983. 1.6
P. L. Butzer e R. J. Nessel. Fourier Analysis and Approximation, volume 1.
Birkh¨auser, Basel, 1971. 1.2, 1.6, 4.6.2
Sergio D. Cabrera e Thomas W. Parks. Extrapolation and spectral estimation with iterative weighted norm modification. IEEE Trans. Signal Processing, 39(4):842– 851, 1991. (document)
James A. Cadzow. An extrapolation procedure for band-limited signals. IEEE Trans.
Acoust., Speech, Signal Processing, 27(1):4–12, 1979. (document), 4.7
James A. Cadzow. Observations on the extrapolation of a band-limited signal pro- blem. IEEE Trans. Acoust., Speech, Signal Processing, 29(6):1208–1209, 1981. (do- cument), 4.7
L. L. Campbell. Sampling theorem for the Fourier transform of a distribution with bounded support. SIAM J. Appl. Math., 16(3):626–636, 1968. 2.3.1, 2.4
Christodoulos C. Chamzas e Wen Yuan Xu. An improved version of Papoulis- Gerchberg algorithm on band-limited extrapolation. IEEE Trans. Acoust., Speech,
Signal Processing, 32(2):437–440, 1984. (document)
K. Chandrasekharan. Classical Fourier Transforms. Springer, Berlin, 1989. 1.2, 1.4, 1.6
C.-Y. Chao. On a type of circulants. Linear Algebra Appl., 6:241–248, 1973. 3.3.1, 3.10
James W. Cooley. How the FFT gained acceptance. IEEE Sig. Proc. Mag., 9(1): 10–13, 1992. 2.4
R. Courant e D. Hilbert. Methods of Mathematical Physics, volume 1. Interscience Publishers, Inc., New York, 1953. Reimpresso: John Wiley & Sons, Wiley Classics Library Edition, 1989. 1.1, 2.1.2, 2.2.3, 4.5, 4.7
P. J. Davis. Interpolation and Approximation. Blaisdell Publishing Company, Waltham, Massachusetts, 1963. Reimpresso: Dover Publications, 1975. (docu- ment), 3.10
P. J. Davis. Circulant Matrices. John Wiley & Sons, New York, 1979. Reimpresso: Chelsea Publishing Company, New York, 1994. 3.7, 3.10
N. Dunford e J. T. Schwartz. Linear Operators Part I: General Theory, volume 1. John Wiley & Sons, New York, 1957. Reimpresso: John Wiley & Sons, Wiley Classics Library Edition, 1988. 4.6.2, 5.6
N. Dunford e J. T. Schwartz. Linear Operators Part III: Spectral Operators, volume 3. John Wiley & Sons, New York, 1971. Reimpresso: John Wiley & Sons, Wiley Classics Library Edition, 1988. 4.6.2
P. J. S. G. Ferreira. Reconstru¸c˜ao de sinais a partir de informa¸c˜ao parcial. Trabalho de s´ıntese, Provas de Aptid˜ao Pedag´ogica e Capacidade Cient´ıfica, Departamento de Electr´onica e Telecomunica¸c˜oes, Universidade de Aveiro, 1988. (document), 3.6.2, 3.10, 4.7
P. J. S. G. Ferreira. Fast iterative reconstruction of distorted ECG signals. Em
Proceedings of the 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, volume II, p´aginas 777–778, Paris, France, 1992a.
(document)
P. J. S. G. Ferreira. Incomplete sampling series and the recovery of missing samples from oversampled band-limited signals. IEEE Trans. Signal Processing, 40(1):225– 227, 1992b. (document), 2.4
REFER ˆENCIAS BIBLIOGR ´AFICAS 131
P. J. S. G. Ferreira. Iterative interpolation of ECG signals. Em Proceedings of the
14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, volume VI, p´aginas 2740–2741, Paris, France, 1992c. (document)
P. J. S. G. Ferreira. A unified approach to a class of constrained interpolation, ex- trapolation and sampling problems. Em Proceedings of RecPad92, 4th Portuguese
Conference on Pattern Recognition, p´aginas 273–279, Coimbra, Portugal, 1992d.
(document)
P. J. S. G. Ferreira. New fast block-iterative algorithms for band-limited interpolation and extrapolation. Em Proceedings of RecPad93, 5th Portuguese Conference on
Pattern Recognition, p´aginas 9–16, Maia, Portugal, 1993. (document), 6.2
P. J. S. G. Ferreira. A group of permutations that commute with the discrete Fourier transform. IEEE Trans. Signal Processing, 42(2):444–445, 1994a. (document) P. J. S. G. Ferreira. Interpolation and the discrete Papoulis-Gerchberg algorithm.
IEEE Trans. Signal Processing, 42(10):2596–2606, 1994b. (document), 3.6.2, 3.6.2,
3.9, 6.2
P. J. S. G. Ferreira. Noniterative and faster iterative methods for interpolation and extrapolation. IEEE Trans. Signal Processing, 42(11):3278–3282, 1994c. (docu- ment)
R. W. Gerchberg. Super resolution through error energy reduction. Opt. Acta, 21(9): 709–720, 1974. (document), 4.2.3, 4.7
P. R. Halmos. Measure Theory. D. Van Nostrand Company, New York, segunda edi¸c˜ao, 1951. 1.6
G. H. Hardy. On an integral equation. Proc. Lond. Math. Soc., II. Ser., 7:444–472, 1909. 4.7
G. H. Hardy. A Mathematician’s Apology. Cambridge University Press, Cambridge, segunda edi¸c˜ao, 1967. 1.6
G. H. Hardy e E. C. Titchmarsh. Solutions of some integral equations considered by Bateman, Kapteyn, Littlewood, and Milne. Proc. Lond. Math. Soc., II. Ser., 23: 1–26, 1924. 4.7
G. H. Hardy e E. C. Titchmarsh. Additional note on certain integral equations. Proc.
Lond. Math. Soc., II. Ser., 30:95–106, 1929. 4.7
J. R. Higgins. Five short stories about the cardinal series. Bull. Am. Math. Soc., New
H. Hochstadt. Integral Equations. John Wiley & Sons, New York, 1973. Reimpresso: John Wiley & Sons, Wiley Classics Library Edition, 1989. 2.2.3, 4.5, 4.7, 5.6 R. A. Horn e C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge,
1990. 1.1
A. S. Householder. The Theory of Matrices in Numerical Analysis. Dover Publications, New York, 1975. 1.1, 2.4
Anil K. Jain e Surendra Ranganath. Extrapolation algorithms for discrete signals with application in spectral estimation. IEEE Trans. Acoust., Speech, Signal Processing, 29(4):830–845, 1981. (document), 4.7
A. J. Jerri. On the application of some interpolating functions in physics. Journal
of Research of the National Bureau of Standards — B, Mathematical Sciences, 73B
(3):241–245, 1969. 2.4
A. J. Jerri. Sampling expansion for a Laguerre-Lα
ν transform. Journal of Research
of the National Bureau of Standards — B, Mathematical Sciences, 80B(3):415–418,
1976. 2.4
A. J. Jerri e D. W. Kreisler. Sampling expansions with derivatives for finite Hankel and other transforms. SIAM Journal on Mathematical Analysis, 6(2):262–267, 1975. 2.4
Abdul J. Jerri. The Shannon sampling theorem — its various extensions and appli- cations: a tutorial review. Proc. IEEE, 65(11):1565–1596, 1977. 1.4, 1.4, 1.6, 2.1, 2.2.3, 2.4
M. C. Jones. The discrete Gerchberg algorithm. IEEE Trans. Acoust., Speech, Signal
Processing, 34(3):624–626, 1986. (document)
Aggelos K. Katsaggelos e Serafim N. Efstratiadis. A class of iterative signal restoration algorithms. IEEE Trans. Acoust., Speech, Signal Processing, 38(5):778–786, 1990. (document)
Dean P. Kolba e Thomas W. Parks. Optimal estimation for band-limited signals inclu- ding time domain considerations. IEEE Trans. Acoust., Speech, Signal Processing, 31(1):113–122, 1983. (document)
T. W. K¨orner. Fourier Analysis. Cambridge University Press, Cambridge, 1988. 1.6 H. P. Kramer. A generalized sampling theorem. J. Math. Physics, 38:68–72, 1959.
1.4, 1.6, 2.4
E. Kreyszig. Introductory Functional Analysis with Applications. John Wiley & Sons, New York, 1978. Reimpresso: John Wiley & Sons, Wiley Classics Library Edition, 1989. 4.6.1, 4.6.1, 5.1, 5.1, 5.4, 5.6
REFER ˆENCIAS BIBLIOGR ´AFICAS 133
H. J. Landau. On the recovery of a band-limited signal, after instantaneous compan- ding and subsequent band limiting. Bell Syst. Tech. J., 39:360–364, 1960. (docu- ment), 1.6, 4.7
H. J. Landau e W. L. Miranker. The recovery of distorted band-limited signals. J.
Math. Anal. Appl., 2:97–104, 1961. (document), 1.6, 4.7
H. J. Landau e H. O. Pollak. Prolate spheroidal wave functions, Fourier analysis and uncertainty – II. Bell Syst. Tech. J., 40:65–84, 1961. (document), 1.3, 4.7
B. F. Logan. On the eigenvalues of a certain integral equation. SIAM J. Math. Anal., 15(4):712–717, 1984. 4.7
B. F. Logan, Jr. Theory of analytic modulation systems. Bell Syst. Tech. J., 57(3): 491–576, 1978. (document)
R. J. Marks II. Introduction to Shannon Sampling and Interpolation Theory. Springer, Berlin, 1991. 1.4, 1.6, 2.4
Robert J. Marks II. Restoring lost samples from an oversampled band-limited signal.
IEEE Trans. Acoust., Speech, Signal Processing, 31(3):752–755, 1983. (document),
2.1, 2.1.2, 2.1.4, 2.4, 4.7
Robert J. Marks II e Tonya Reightley. On iterative evaluation of extrema of integrals of trigonometric polynomials. IEEE Trans. Acoust., Speech, Signal Processing, 33 (4):1039–1049, 1985. (document), 4.7
F. A. Marvasti. An iterative method to compensate for the interpolation distortion.
IEEE Trans. Acoust., Speech, Signal Processing, 37(10):1617–1621, 1989. (docu-
ment), 4.7
F. A. Marvasti, M. Analoui, e M. Gamshadzahi. Recovery of signals from nonuniform samples using iterative methods. IEEE Trans. Signal Processing, 39(4):872–878, 1991. (document), 4.7
Elias Masry. The recovery of distorted band-limited stochastic processes. IEEE Trans.
Inform. Theory, 19(4):398–403, 1973. (document)
F. J. Murray. On complementary manifolds and projections in spaces Lp and lp.
Trans. Amer. Math. Soc., 41:138–152, 1937. 5.2
P. S. Naidu e Bina Paramasivaiah. Estimation of sinusoids from incomplete time series.
IEEE Trans. Acoust., Speech, Signal Processing, 32(3):559–562, 1984. (document),
4.7
H. Nyquist. Certain topics in telegraph transmission theory. Trans. AIEE, 47:617–644, 1928. 1.6
A. Papoulis. A new algorithm in spectral analysis and band-limited extrapolation.
IEEE Trans. Circuits Syst., 22(9):735–742, 1975. (document), 4.2.3, 4.7
A. Papoulis. Signal Analysis. McGraw-Hill International Editions, New York, 1987. 1.6, 2.2.3, 3.5
Athanasios Papoulis e Christodoulos Chamzas. Detection of hidden periodicities by adaptive extrapolation. IEEE Trans. Acoust., Speech, Signal Processing, 27(5): 492–500, 1979. (document)
H. R. Pitt. Measure and Integration for Use. Oxford University Press, Oxford, 1987. 1.6
W. Pogorzelski. Integral Equations and Their Applications. Pergamon Press, Oxford, 1966. 4.7
L. C. Potter e K. S. Arun. Energy concentration in band-limited extrapolation. IEEE
Trans. Acoust., Speech, Signal Processing, 37(7):1027–1041, 1989. (document)
W. H. Press, B. P. Flannery, S. A. Teukolsky, e W. T. Vetterling. Numerical Recipes
in C. Cambridge University Press, Cambridge, 1988. (document)
M. D. Rawn. On nonuniform sampling expansions using entire interpolation functions, and on the stability of Bessel-type sampling expansions. IEEE Transactions on
Information Theory, 35(3):549–557, 1989. 1.6
F. Riesz e B. Sz.-Nagy. Functional Analysis. Frederick Ungar Publishing Co., New York, 1955. Reimpresso: Dover Publications, 1990. 4.7, 5.6
H. E. Rose. A Course in Number Theory. Oxford University Press, Oxford, 1988. 3.3.1, 3.3.1, 3.3.1
W. Rudin. Real and Complex Analysis. McGraw-Hill International Editions, New York, terceira edi¸c˜ao, 1987. 1.3, 1.6, 5.6
M. S. Sabri e W. Steenaart. An approach to band-limited signal extrapolation: The extrapolation matrix. IEEE Trans. Circuits Syst., 25(2):74–78, 1978. (document), 4.7
M. Shaker Sabri e Willem Steenaart. Comments on “An extrapolation procedure for band-limited signals”. IEEE Trans. Acoust., Speech, Signal Processing, 28(2):254, 1980. (document), 4.7
M. Shaker Sabri e Willem Steenaart. Rebuttal to “Observations on the extrapolation of a band-limited signal problem”. IEEE Trans. Acoust., Speech, Signal Processing, 29(6):1209, 1981. (document), 4.7
REFER ˆENCIAS BIBLIOGR ´AFICAS 135
I. W. Sandberg. On the properties of some systems that distort signals — I. Bell
Syst. Tech. J., 42:2033–2046, 1963. (document), 4.7
J. L. C. Sanz e T. S. Huang. Unified Hilbert space approach to iterative least-squares linear signal restoration. J. Opt. Soc. Am., 73(11):1455–1465, 1983a. (document), 5.6
Jorge L. Sanz e Thomas S. Huang. Iterative time-limited signal reconstruction. IEEE
Trans. Acoust., Speech, Signal Processing, 31(3):643–649, 1983b. (document)
Jorge L. C. Sanz e Thomas S. Huang. Discrete and continuous band-limited signal extrapolation. IEEE Trans. Acoust., Speech, Signal Processing, 31(5):1276–1285, 1983c. (document)
Jorge L. C. Sanz e Thomas S. Huang. Some aspects of band-limited signal extrapo- lation: Models, discrete approximations, and noise. IEEE Trans. Acoust., Speech,
Signal Processing, 31(6):1492–1501, 1983d. (document)
Jorge L. C. Sanz e Thomas S. Huang. A unified approach to noniterative linear signal restoration. IEEE Trans. Acoust., Speech, Signal Processing, 32(2):403–409, 1984. (document), 4.7
K. D. Sauer e J. P. Allebach. Iterative reconstruction of band-limited images from nonuniformly spaced samples. IEEE Trans. Circuits Syst., 34(12):1497–1506, 1987. (document), 4.7
R. W. Schafer, R. M. Mersereau, e M. A. Richards. Constrained iterative restoration algorithms. Proc. IEEE, 69(4):432–450, 1981. (document), 3.6.2, 3.10, 4.7
Heinz-Josef Schlebusch e Wolfgang Splettst¨oßer. On a conjecture of J. L. C. Sanz and T. S. Huang. IEEE Trans. Acoust., Speech, Signal Processing, 33(6):1628–1630, 1985. (document)
S. Singh, S. N. Tandon, e H. M. Gupta. An iterative restoration technique. Sig. Proc., 11:1–11, 1986. (document)
D. Slepian. Prolate spheroidal wave functions, Fourier analysis and uncertainty — V: The discrete case. Bell Syst. Tech. J., 57(5):1371–1430, 1978. 2.1.2, 2.1.4
D. Slepian e H. O. Pollak. Prolate spheroidal wave functions, Fourier analysis and uncertainty — I. Bell Syst. Tech. J., 40(1):43–63, 1961. (document), 1.3, 4.7 D. R. Smart. Fixed Point Theorems. Cambridge University Press, Cambridge, 1980.
3.10, 4.7
D. E. Smith. A Source Book in Mathematics. McGraw-Hill Book Company, Inc., New York, 1929. Reimpresso: Dover Publications, 1959. 2.4
F. Smithies. Integral Equations. Cambridge University Press, Cambridge, quarta edi¸c˜ao, 1970. 4.5, 4.7
A. Sobczyk. Projections in Minkowsky and Banach spaces. Duke Math. J., 8:78–106, 1941. 5.2
J. M. Sousa Pinto. Sampling Expansions and Generalized Translation Invariant Li-
near Systems. Tese de Doutoramento, Dept. of Mathematics, Cranfield Institute of
Technology, U.K., 1983. 1.6
J. Spanier e K. B. Oldham. An Atlas of Functions. Hemisphere Publishing Corpora- tion, Washington, 1987. 2.2.3
G. W. Stewart. Review of matrix computations by Gene H. Golub and Charles F. Van Loan. Linear Algebra and its Applications, 95:211–215, 1987. 2.4
Barry J. Sullivan e Bede Liu. On the use of singular value decomposition and de- cimation in discrete-time band-limited signal extrapolation. IEEE Trans. Acoust.,
Speech, Signal Processing, 32(6):1201–1212, 1984. (document)
E. C. Titchmarsh. Introduction to the Theory of Fourier Integrals. Oxford University Press, Oxford, segunda edi¸c˜ao, 1948. 1.6, 2.4, 4.6.1, 4.6.2, 4.7
E. C. Titchmarsh. The Theory of Functions. Oxford University Press, Oxford, segunda edi¸c˜ao, 1968. 1.6, 4.6.2
Victor T. Tom, Thomas F. Quatieri, Monson H. Hayes, e James H. McClellan. Con- vergence of iterative nonexpansive signal reconstruction algorithms. IEEE Trans.
Acoust., Speech, Signal Processing, 29(5):1052–1058, 1981. (document), 3.10
F. G. Tricomi. Integral Equations. Dover Publications, New York, 1985. 4.5, 4.7 R. S. Varga. Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, New Jersey,
1962. 2.1.4, 2.1.5, 2.4, 3.8, 3.8.4
G. A. Viano. On the extrapolation of optical image data. J. Math. Physics, 17(7): 1160–1165, 1976. (document)
P. Weiss. Sampling theorems associated with Sturm-Liouville systems. Bulletin of the
American Mathematical Society, 63:242, 1957. 1.6
E. T. Whittaker. On the functions which are represented by the expansions of the interpolation-theory. Proceedings of the Royal Society of Edinburgh, 35:181–194, 1915. 1.6
N. Wiener. The Fourier Integral and Certain of Its Applications. Cambridge Univer- sity Press, Cambridge, 1933. Reimpresso: Dover Publications, 1958. 1.6
REFER ˆENCIAS BIBLIOGR ´AFICAS 137
R. G. Wiley. On an iterative technique for recovery of bandlimited signals. Proc.
IEEE, 66(4):522–523, 1978a. (document)
R. G. Wiley. Recovery of bandlimited signals from unequally spaced samples. IEEE
Trans. Commun., 26(1):135–137, 1978b. (document), 4.7
R. G. Wiley. Concerning the recovery of a bandlimited signal or its spectrum from a finite segment. IEEE Trans. Commun., 27(1):251–252, 1979. (document), 4.7 R. G. Wiley, H. Schwarzlander, e D. D. Weiner. Demodulation procedure for very
wide-band FM. IEEE Trans. Commun., 25(3):318–327, 1977. (document), 4.7 J. H. Wilkinson. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1978.
2.4
Wen Yuan Xu e Christodoulos Chamzas. On the extrapolation of band-limited func- tions with energy constraints. IEEE Trans. Acoust., Speech, Signal Processing, 31 (5):1222–1234, 1983. (document)
D. C. Youla. Generalized image restoration by the method of alternating orthogonal projections. IEEE Trans. Circuits Syst., 25(9):694–702, 1978. (document), 4.7 D. M. Young. Iterative Solution of Large Linear Systems. Academic Press, Orlando,
terceira edi¸c˜ao, 1971. 2.1.4, 2.1.5, 2.4, 3.8, 5.6
X.-W. Zhou e X.-G. Xia. The extrapolation of high-dimensional band-limited func- tions. IEEE Trans. Acoust., Speech, Signal Processing, 37(10):1576–1580, 1989a. (document)
X.-W. Zhou e X.-G. Xia. A Sanz-Huang conjecture on band-limiting signal extrapola- tion with noise. IEEE Trans. Acoust., Speech, Signal Processing, 37(9):1468–1472, 1989b. (document)