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Dym H., McKean H. — Fourier Series and Integrals (Probability & Mathematical Statistics Monograph)
Dym H., McKean H. — Fourier Series and Integrals (Probability & Mathematical Statistics Monograph)



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Íàçâàíèå: Fourier Series and Integrals (Probability & Mathematical Statistics Monograph)

Àâòîðû: Dym H., McKean H.

Àííîòàöèÿ:

The ideas of Fourier have made their way into every branch of mathematics and mathematical physics, from the theory of numbers to quantum mechanics. Fourier Series and Integrals focuses on the extraordinary power and flexibility of Fourier's basic series and integrals and on the astonishing variety of applications in which it is the chief tool. It presents a mathematical account of Fourier ideas on the circle and the line, on finite commutative groups, and on a few important noncommutative groups. A wide variety of exercises are placed in nearly every section as an integral part of the text.


ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1972

Êîëè÷åñòâî ñòðàíèö: 301

Äîáàâëåíà â êàòàëîã: 17.04.2014

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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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 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, in a Hilbert space      24 26 80 170 255 270 see "Span" "Unit-perpendicular
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
Boerner, 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 $L^{1}$      37 38
Convergence in $L^{2}$      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
D'Alembert      1
d'Alembert, formula of      2 71 109 139
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
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
Erdos      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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