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Index
A(n, d) 2,19 A(n, θ) 2,56
A∗≡conjugated transpose of matrixA AÌ≡transpose of matrixA
α(G)≡stability number
αm(G)≡measurable stability number Andrews,G.E. 35,36,38,42,43,45,
46,60,69
Askey,R. 35,36,38,42,43,45,46,60, 69
association scheme 20
Aut(G) ≡automorphism group of a graph
Axiom of Choice
relation to the chromatic number of the Euclidean space 7
Axler,S. 60 Bachoc,C. 23
Bessel function of the first kind 69 Bessel’s equation for the∼ 69 integral representation 69 interlacing property 71 properties of the∼ 69–72 zeros of the∼ 39,70,71 Bessel,F.W. 70
binary code 2,19
linear programming bound 5,20 Bochner’s theorem 87
Bochner,S. 26 Bourdon,P. 60 Bourgain,J. 80 Bruijn,N.G.de 7,24
Bukh,B. 80,81
C+y≡translation of a set Ckλ≡polynomial,ultraspherical χ(G)≡chromatic number
χ(Rn) ≡chromatic number of the Eu-clidean space
χm(Rn)≡measurable chromatic number of the Euclidean space
chromatic number 1,11
bounds for the∼of the plane 6 bounds for the∼ofR3 6 computing the∼ 11
measurable∼of Euclidean space 6,24 measurable∼of graphs on the
sphere 24
measurable∼of the Euclidean space, asymptotic upper bound 8 measurable∼of the Euclidean space,
exponential growth 40,73 measurable∼of the Euclidean space,
upper bounds 8
measurable∼,relation to measurable stability number 24
of graphs on the sphere 24
of graphs on the sphere,relation to mea-surable∼ 24
of the Euclidean space 5,24
of the Euclidean space,asymptotic upper bound 7
of the Euclidean space,exponential growth 6
105
106 Index
of the Euclidean space,relation to mea-surable∼ 6,8
of the Euclidean space,relation to the Axiom of Choice 7
of the unit-distance graph 3 relation to stability number 11 color class (in a coloring) 11 Coulson,D. 6,7
Delsarte,P. 5,10,20,56,57 δ(C)≡upper density distance graph 2
on the Euclidean space 84 on the unit sphere 23 distance-avoiding set 9,63
maximum density of ∼ 3,6,9,63 maximum density of ∼,exponential
de-cay 9,73
restriction to periodic sets 67 eigenfunction of a kernel 26 eigenvalue of a kernel 26 Erd˝os,P. 6,7,24,80 Euclidean space
bounds for the chromatic number of the plane 6
bounds for the chromatic number ofR3 6
chromatic number of the∼ 5,24 chromatic number of the∼,asymptotic
upper bound 7
chromatic number of the∼,exponential growth 6
chromatic number of the∼,relation to measurable chromatic number 6 chromatic number of the∼,relation to
the Axiom of Choice 7
chromatic number of ∼,relation to mea-surable chromatic number 8 coloring of the plane 6
distance graph on the∼ 84 kissing number of the plane 2 kissing number of the∼ 2 measurable chromatic number of
the∼ 6,24
measurable chromatic number of the∼, asymptotic upper bound 8
measurable chromatic number of the∼, exponential growth 40
measurable chromatic number of the∼, upper bounds 8
norm of a vector in∼(x) 13 standard inner product in∼(x·y) 13 theta number for graphs on the∼ 85–
89
unit-distance graph on the∼ 3,6 Falconer,K.J. 6–8,24,40,49,74,77,
80,81 Feldheim,E. 38 Fourier coefficient 67 Frankl,P. 6,9 function
almost periodic∼ 67 autocorrelation∼ 67
autocorrelation∼,expansion 68,85 Fourier coefficient of a periodic∼ 67 of positive type 85
over the cube[−R, R]n 25 periodicity lattice of a∼ 66 periodic∼ 65
pole of zonal spherical∼ 32 radial 93
square-integrable periodic∼ 66 zonal spherical∼ 32
zonal spherical∼,properties 32 Furstenberg,H. 10,80
Gerard,B. 48
Goethals,J.M. 10,56,57 Gr¨otschel,M. 14 graph 11
adjacency in a∼ 11
automorphism group of a∼ 12 automorphism of a∼ 12 coloring of a∼ 11 complementary graph 11 distance∼ 2
distance∼on the unit sphere 23 edge set 11
finite∼ 11 Moser∼ 6
stable set in a∼ 2,11 subgraph of a∼ 11 unit-distance∼ 3,6 vertex set 11
vertex-transitive∼ 12 Gvozdenovi´c,N. 15
Index 107 Hn≡Hamming cube
Hadwiger,H. 6 Halmos,P.R. 30 Hamming cube 2,19
invariant matrix over the∼, characteriza-tion 20
isometry group of the∼ 19 Hamming distance 2,19 harmonic analysis onRn 65–67 Hilbert space 25
Hilbert-Schmidt theorem 26 hypergeometric series 69 inner product
invariance of ∼onL2(Sn−1) un-derO(Rn) 29
inL2(Sn−1×Sn−1)(A, B) 25 inL2(Sn−1)((f, g)) 25
of periodic functions (f, g) 66 onL2([−1,1])((f, g)α,β) 30 standard Euclidean∼(x·y) 13 trace∼of matrices (A, B) 13 invariant integration 30
Isbell,J. 6
Jα≡Bessel function of the first kind Kabatyanskii,G.A. 10,56,57 Karp,R.M. 11
Katznelson,Y. 10,65,67,80 kernel 25
as linear operator 25 condition for positivity 26 eigenfunction of a∼ 26 eigenvalue of a∼ 26 Hilbert-Schmidt theorem 26 Hilbert-Schmidt∼ ≡kernel Mercer’s theorem 27
norm of a∼over the sphere (A) 25 over the sphere,characterization of
posi-tive,invariant∼ 30,31,57 over the sphere,invariant under
O(Rn) 29
over the sphere,symmetrization of a∼ 30
positive 25
positivity in terms of eigenvalues 26 symmetric 25
trace of a∼ 27 trace-class∼ 27
translation invariant∼inRn 88 kissing number 2
Knuth,D.E. 21 L2(Rn/Λ) 66 l2(t),computing∼ 35 ln(t) 33
complete characterization 47 computing∼ 37
partial characterization 41,42 Larman,D.G. 7,8,97 lattice
dual∼ 66
fundamental domain of a∼ 66 Levenshtein,V.I. 10,56,57 linear programming 13
relation with semidefinite program-ming 13
Lov´asz,L. 4,14,15,17,18,22,24,30 m1(Rn)≡maximum density of
distance-avoiding set MacWilliams,F.J. 19 Marstrand,J.M. 80 matrix
conjugate transpose of a∼ 12 entry notation 12
Hermitian∼ 12
indexed by a finite set 12
invariant∼overHn,characterization 20 invariant∼under automorphism
group 18
invariant∼under the isometry group ofHn 20
positive definite∼ 14 positive semidefinite∼ 12 symmetric∼ 12
trace of a∼ 12 transpose of a∼ 12 Mattila,P. 30,56,76 McEliece,R.J. 10,19,20,56 Mercer’s theorem 27
Moser,L. 6 Moser,W. 9 Nebe,G. 23 Nechushtan,O. 6,8 Nelson,E. 6 Nemirovskii,A. 14 Nesterov,Y. 14
108 Index
norm inL2(Sn−1)(f) 25 Oliveira Filho,F.M.de 23,63 ω≡surface measure on the sphere Ωn 64
asymptotic behavior 72 derivative of ∼ 70 global minimum 71 positive extrema 70,71 optimization
optimal solution of an∼problem 14 constraints of an ∼problem 13 dealing with numerical approximation
in∼ 78
exploiting symmetry in∼ 5
feasible solution of an∼problem 14 feasible∼problem 14
infeasible∼problem 14
objective function of an ∼problem 13 objective value of a solution 14 optimal value of an∼problem 14 terminology for ∼problems 13 use of supremum/infimum in∼
prob-lems 14
variables of an ∼problem 13 O(Rn)≡orthogonal group orthogonal group 29
Haar measure over the ∼ 29 invariance of surface measure on the
sphere under the∼ 29 P(α,β)k 31
Pk(α,β)≡Jacobi polynomial Parserval’s formula 67 polynomial
inner product Jacobi∼s ((f, g)α,β) 30 Jacobi∼ 30
Jacobi∼s,Askey’s conjecture 48 Jacobi∼s,complete orthogonal system
forL2([−1,1]) 31
Jacobi∼s,interlacing property of 41 Jacobi∼s,normalization 31 Jacobi∼s,open questions 46–49 Jacobi∼s,weight function for 30 orthogonal∼s 30
ultraspherical∼s 38 Quas,A. 80
Raigorodskii,A.M. 7,9 Ramey,W. 60
Reed,M. 25,87 regular measure 27 Riesz,F. 25,59 Roberts,L. 48
Rodemich,E.R. 10,19,20,56 Rogers,C.A. 7,8,97
Roy,R. 35,36,38,42,43,45,46,60, 69
Rumsey,Jr.,H.C. 10,19,20,56 Sn−1≡sphere,unit
Schoenberg,I.J. 30,31,57,70,93 Schrijver,A. 10,13,14,19–21,56 Seidel,J.J. 10,56,57
semidefinite programming 13 complexity of∼ 14 dual of a∼problem 14 duality in∼ 14
ellipsoid method applied to∼ 14 interior-point algorithms for∼ 14 problem 4,13
relation with linear programming 13 strict feasibility of a∼problem 14,16 strong duality in∼ 14
weak duality in∼ 14 Simon,B. 25,87 Sloane,N.J.A. 19 Soifer,A. 6,7 sphere,unit 2,23
chromatic number of graphs on the∼ 24
complements of graphs on the∼ 52 distance graphs on the∼ 23 inner product for functions on the∼
((f, g)) 25
measurable chromatic number of graphs on the∼ 24
measurable stability number of graphs on the∼ 24
norm of a kernel over the∼(A) 25 norm of function over the∼(f) 25 properties of the theta number for
graphs on the∼ 52–55 regularity of surface measure on
the∼ 27
surface measure on the∼(ω) 3,23 theta number for graphs on the∼ 27–
29
Index 109 theta number of complements of graphs
on the∼ 53 spherical code 2,56
linear programming bound for∼s 56 stability number 1,2,11
computing the∼ 4,11 measurable∼ 3
measurable∼of graphs on the sphere 24
measurable∼,relation to measurable chromatic number 24
relation to chromatic number 11 Steinhaus’ theorem 81
generalization by Weil 49 Stromberg,K. 49
surface measure on the sphere (ω) 3,23 regularity of the∼ 27
invariance under the orthogonal group 29
Sz´ekely,L.A. 6,8,9,40,74,77,80 Sz.-Nagy,B. 25,59
Szeg¨o,G. 30,36–38,41,45,48 theta number 4,15
as lower bound for chromatic num-ber 16
as upper bound for stability number 15 computational complexity 21
dual formulation for the∼of a graph 16
for compact regular measure spaces 29 for complements of graphs on the
sphere 53
for graphs on the Euclidean space 85–
89
for graphs on the sphere 27–29 for graphs on the sphere,expression 34 monotonicity of the∼ 17
properties of the∼ 17
properties of the∼for graphs on the sphere 52–55
strengthening of the ∼(ϑ) 19 ϑparameter 19
ϑ(G)≡theta number topological group 29 TrA≡trace of matrixA tradition 26
translation of a set 67
unit-distance graph 3,6 upper density 3,63 Vallentin,F. 23,63 vector
column∼ 12
Euclidean norm of a∼(x) 13 indexed by a finite set 12 Vilenkin,N.J. 38
Watson,G.N. 69–72
weight function for Jacobi polynomials 30 Weil,A. 49
Weiss,B. 10,80 Wilson,R.M. 6,9 Wong,R. 45,48
Wormald,N.C. 8,40,74,77 Zhang,J.-M. 45,48