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| Dym H., McKean H.P. — Fourier Series and Integrals |
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| Ïðåäìåòíûé óêàçàòåëü |
Addition formula 249—251 262 269—270 279
Affine transformation 263
Akhiezer, N.I. 14 29 80 194 196
Algebra see "Convolution algebra"
Almost everywhere (a.e.) 10
Analytic functions 144—156
Anderson, W.N., Jr. 221
Annihilator 28
Antisymmetric function 134
Approximate identity 105
Arithmetic sequence, distribution of 54—56
Arthurs, E. 131
Associated Bessel function 268—269
Associated conical function 279
Associated Legendre polynomial 253—254
Auslander, L. 273
Axis 229—232
Bachman, P. 223
Band-limited function 121—132
Basis in a Hilbert space 24 26 80 170 255 270 see "Span" "Unit-perpendicular
Basis of eigenfunctions (of Sturm — Liouville operator) 73
Basis of Haar functions 27—28
Basis of Hermite functions 98
Basis of Laguerre functions 153 165
Basis of Legendre polynomials 36 246—247
Basis of prolate spheroidal wave functions 123
Basis of sampling functions 131
Basis of trigonometric functions 30 57 60 81
Basis of Walsh functions 272
Basis theorem, for finite commutative groups 215—217
Basis, of skew matrix 231
Bateman, H. 106
Bellman, R. 54 225
Berberian, S.K. 29
bernoulli 2
Bessel equation 133
Bessel functions 132—133 143 266—267
Bessel inequality 25 27
Bessel transform 132—133 267—268
Beurling, A. 169
Birkhoff, G. 74 80 203
Bochner, S. 282
Boemer, H. 228 256
Borel measurable set 7
Borel — Cantelli lemma 11 19
Cantor set 8
Caratheodory, C. 273
Carleman, T. 169 282
Carleson, L. 34
Carslaw, H.S. 43 44 282
Case, K.M. 184
Cauchy formula 148—151 167
Cauchy theorem 145—148
Cauchy — Riemann equations 148
Central limit theorem 114—116 142
Central subgroup 274
Chandrasekhar, S. 142
Chandrasekharan, K. 141 222 282
Character of group 205—206 210 218—220 222 223 226—227 229 232 241 249
Character of representation 258—260
Chebyshev’s inequality 11 19
Class function 258
Closed curve 145
Closed subspace 28
Colatitude 233 239
compact function x 21 37
complete 13 16 37
Conical functions 277
Conjugacy class 258
Convergence in  37 38
Convergence in  16 18
Convergence, pointwise a.e. 16—18
Convolution algebras 41—43 87—88 103 212—213 237—239 262 264 277
Coppel, W.A. 2 282
Copson, E.T. 144
Corner 145
Coset 204—205
Coset, double 233 262 263 275
Coset, group of 214—215 220
Coset, space of 232—233 262 263 275
Cosine integral 92—93 132 134 262—263
Cosine series 60
Courant, R. ix 49 74 131 196
Crimmins, T. 221
de Branges, L. 194
de Hoffman, F. 184
de la Vallee-Poussin, C. 5
De Moivre 115
Dense 13 21 30 34
Descent, method of 136—137
Dimension, of Hilbert space 22 26
Dini’s test 40 105
Direct product, of groups 215
Directed curve 145
Dirichlet 2 105
Dirichlet kernel 31—35 40 41 45 260
Divergence theorem 147—148
Doetsch, G. 213
Domain 144
Dominated Convergence Theorem 10
Double coset 233 262 263 275
Dual group 206 207 210 218—222
Duren, P. 169 194
D’Alembert 1
d’Alembert, formula of 2 71 109 139
Edwards, R.E. 42 43 282
Eigenfunctions (eigenvalues) of Fourier transform 97—98
Eigenfunctions (eigenvalues) of integral operators 123—126 130—131
Eigenfunctions (eigenvalues) of Laplacian 242—245 247 251—252 265—266 268—269 270 277 279
Eigenfunctions (eigenvalues) of ordinary differential operators 56—60 72—80 99 208 211
Eigenvalues see Eigenfunctions"
Energy of string 72
Entire function 154
Erdoes 196
Ergodic 54
Euclidean motion group 261—273
Euler 2
Euler — Maclaurin summation formula 112—114
Exponential type 155 158
Factor group 214—215 220
Fatou’s Lemma 10 19
Fejer kernel 35 38 41
Fejer theorem 34—36 49 55
Fejer, L. 34 103 106
Feller, W. 53 84 115 187 196
Feynman, R. x
Filter 170—176
Finicky proof 41
Flanders, M. 61
Ford, G.W. 54
Fourier Bessel transform 132—133 267—268
Fourier coefficient 23 26 39
Fourier cosine integral 92—93 132 134 262—263
Fourier cosine series 60
Fourier integral 86—105 132—134 160—161 209—212
Fourier integral as a rotation 97—98
Fourier integral, application of 106—116 134—143
Fourier integral, growth of 116—121 156—160
Fourier inversion formula 88 90 94 103 112 132—133
Fourier series 30—46 81—85 206—209
Fourier series on finite commutative groups 217—219
Fourier series on rotation group 256 261
Fourier series, application of 46—72 82—84
Fourier sine integral 92—93 134 263
Fourier sine series 57
Fourier — Legendre series 245—249
Fourier — Mehler transform 278
Fourier, J. 2 282
Fredholm alternative 60
Frobenius 4 227
| Fubini’s Theorem 12 41 102
Fundamental solution, of heat equation 64
Fundamental theorem of algebra 154
Gain 170 172
Gauss 215 222 223
Gauss kernel 104 108 111—112
Gauss lattice point formula 140
Gaussian sum 223—226
Gelfand, I.M. 229 256 261 281
Gibbs phenomenon 43—46
Gibbs, J.N. 43
Ginibre, J. 222
Glazman, I.M. 14 29
Goldberg, R.R. 282
Graev, M.I. 261 281
Graf’s addition formula 269—270
Gram — Schmidt recipe 25 26 153 247
Green function 58 59 60 64 67 72 80
Group interpretation of classical Fourier analysis 206—212
Group, central subgroup of 274
Group, direct product of 215
Group, dual 206 207 210 218—222
Group, factor 214—215 220
Group, finite commutative 214—217
Group, introduction to 203—206
Group, orthogonal 228—229
Group, permutation 134 204—206
Group, representations 227 255—261
Group, representations of rigid motions of Euclidean line 262
Group, representations of rigid motions of Euclidean plane 261 263
Group, representations of rigid motions of hyperbolic plane 273—274
Group, rotation 228—236
Group, special orthogonal 229—236
Group, stability 275
Haar functions 27—28
Hammermesh, M. 256
Hardy function applications 194—196
Hardy function spaces 161—176 187—194
Hardy, G.H. 3 22 282
Hardy, theorem of 156—158
Heat equation 60—70 82 107—109
Heaviside 213
Heisenberg’s inequality 100 116—121 122 156
Heitler, W. 119 251
Helgason, S. 281
Helson, H. 169 194
Hermite function 98—101 157 211
Hewitt, E. 5 282
Hilbert space 13—15 22 26 29 37
Hilbert transform 93
Hilbert, D. 49 74 131
Hobson, E.W. 2 282
Hoffman, K. 169 194
Homogeneous function 254
Homogeneous polynomial 253 255
Homomorphism, convolution algebra 185 208—209 211—213 214 239—241 249 262 264 273
Homomorphism, group-to-group 204—205
Hopf — Wiener factorization 176—184
Hopf, E. 177 184
Horwitz, H. 221
Hurwitz 49
Hyperbolic distance 274
Hyperbolic plane 273—281
Ideal 41
Ikehara, S. 196
Images, Lord Kelvin’s method of 66 67 68 108
Improper integral 95—96 97 150 151
Indicator function—x 9
Infinitely often (i.o.) 11 19
Ingham, A.E. 196
Inner functions (filters) 172 193—194
Inner product 13 14 16 22 27 29 218—219
Integral function 154
Integration on (other continuous) groups 234—236 260 262 264 276
Integration, advantage of Lebesgue theory of 16
Integration, introduction to Lebesgue theory of 5—12
Invariant subspace 207—208 210—211 253
Inverse Fourier transform (formula) 88 90 94 103 112 132—133
Irreducible-representation 227
Isomorphic groups 205 207 210 219 220
Isomorphic Hilbert spaces 22 26
Isomorphism between groups 204 207 210 218
Isomorphism between Hilbert spaces 30 272—273
Isoperimetric problem 49—51
Ito, K. 221
Jacobi’s identity 52—54 65—66 67 85 112
Jensen’s Inequality 168 192—193
John, F. ix 137
Kac, M. 106 211
Katznelson, Y. 282
Kelvin, Lord (method of images) 66 67 68 108
Kingman, J.F.C. 187
Kolmogorov, A.N. 34
Lagrange 2
Laguerre functions 153 165
Landau, H.J. 122 130
Landsberg — Schaar identity 225—226
Laplace 115
Laplace operator 134 242—247 251—253 255 265—266 268—271 276—279
Laplace transform 213
Lardy, L.J. 213
Latitude 233
Lattice, dual 84
Lattice, Gauss’ formula 40
Lattice, Minkowksi’s theorem 140—141
Lattice, nonstandard 84
Lattice, random walk on 82—84
Lattice, standard 81
Lebesgue 2
Lebesgue integration 5—12 16
Lebesgue measurable 7
Leblanc, N. 273
Lee, Y.W. 172
Legendre operator 74
Legendre polynomials 26 242 245—249
Legendre symbol 223
Levinson, N. 144 196
Liouville’s theorem 154 158
Littlewood 3
Local convergence, of Fourier series 39—40
Local-global duality 31 93 106 116
Loomis, L. 282
Ludwig, D. 137
MacLane, S. 203
Maximal invariant subspace 208
Maximum principle for analytic functions 155
Maximum principle for solutions of heat equation 64—65
Maxwell equation 256
Maxwell poles 254—255
Maxwell, J.C. 254
McKean, H.P. 222
Measurable function 9—10 12
Measurable set 7—8 12
Measure 7 8 11 12
Mehler transform 278
Mellin transform 103
Mercer’s theorem 131
Michaelson, A. 44
Milne’s equation (problem) 176—178 181—184
Minimal invariant subspaces 208 253
Minimal type 158
Minkowski’s theorem in the geometry of numbers 140—143
Minkowski’s triangle inequality 16
Minlos, R. 229 256
Moebius function 214 226
Moebius inversion formula 214
Momentum 118
Monotone Convergence Theorem 10 83
Morera’s theorem 150 152
Mueller, C. 253 256
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