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Szego G. — Orthogonal polynomials
Szego G. — Orthogonal polynomials

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Название: Orthogonal polynomials

Автор: Szego G.

Аннотация:

Recent years have seen a great deal of progress in the field of orthogonal
polynomials, a subject closely related to many important branches of analysis.
Orthogonal polynomials are connected with trigonometric, hypergeometric,
Bessel, and elliptic functions, are related to the theory of continued fractions and to important problems of interpolation and mechanical quadrature, and are of occasional occurrence in the theories of differential and integral equations.
In addition, they furnish comparatively general and instructive illustrations of
certain situations in the theory of orthogonal systems. Recently, some of these polynomials have been shown to be of significance in quantum mechanics and in mathematical statistics.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1939

Количество страниц: 431

Добавлена в каталог: 19.07.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel      100 401
Abel's continuity theorem      92
Abel's transformation      2 91 135
Abelsummability      245 273.
Achieser      5 37 42 401 416
Adamoff      203 248 249 250 401
Addition theorems      58 97
Additive operation      11 12
Airy's function      18 19 131 201 234 239 243 377 382.
Antipole condition      246 249 265
Approximation      5 ff
Askey      xi 98 99 110 273 348 363 391 392 416
Askey — Fitch      96 363 416
Askey — Gasper      97 99 110 190 392 417
Askey — Pollard      273 417
Askey — Steinig      97 158 363 417
Askey — Wainger      99 190 273 417
Associated function, Legendre      84
Asymptotic formula of Bessel functions      15 16
Asymptotic formula of classical polynomials      191 ff
Asymptotic formula of general orthogonal polynomials      v vi 296
Asymptotic formula of Hermite polynomials      132 191 194 198 218 235 242
Asymptotic formula of Jacobi functions of second kind      201 225
Asymptotic formula of Jacobi of Hilb's type      169 197 202 214
Asymptotic formula of Jacobi of Mehler — Heine type      167 192 202.
Asymptotic formula of Jacobi polynomials      58 167 168 191 201 202 212 225
Asymptotic formula of kernel polynomials      369 ff
Asymptotic formula of Laguerre polynomials      132 191 193 194 198 236 237 240
Asymptotic formula of Laguerre polynomials of Hilb's type      177 199 203 216 219 384
Asymptotic formula of Laguerre polynomials of Mehler — Heine type      193 194 217
Asymptotic formula of Laguerre polynomialsof Plancherel — Rotach type      200 201 227 Perron's
Asymptotic formula of Legendre functions of second kind      198 212 222
Asymptotic formula of Legendre polynomials      191 193 194 243
Asymptotic formula of Legendre polynomials of Hilb's type      195 202 212 Laplace
Asymptotic formula of polynomials orthogonal on a curve      371 ff
Asymptotic formula of polynomials orthogonal on the unit circle      297 ff
Asymptotic formula of ultraspherical polynomials      196. 197 206 208 209 212
Bailey      273 392 417
Balazs — Turan      347 417
Banach      13 401
Bateman      96 98 99 273 417
Bateman Project      ix 95 110 243 393 394 417
Bernstein's theorem on trigonometric polynomials      5 280 304.
Bernstein, S.      vi 9 31 42 159 165 168 296 299 300 303 314 315 330 401
Bessel functions      v 14 95 96 102 126 140 166 167 191 193 202 203 212 225 243 272 353 362 363 380 384 387 392 400. Zeros
Bessel's inequality      25 38 289 316 322 333 367
Blumenthal      268 310 401
Bochner      108 273 401 417
Bonami — Clerc      273 417
Bottema      132 401
Bounded variation, functions of      12
Brauer, A.      131 401
Bruns      122 125 136 138 401
Buell      124 ff. 401
Butlewski      166 417
Capacity      see Transfinite diameter
Caratheodory      364 402.
Carleman      366 402
Carleson      273 417
Cauchy — Hadamard formula      248 253 312
Cauchy's principle value      278
Cauchy's theorem      69 106 221 275 289 324 370 372
Centroid      189
Cesaro summability      vi 244 246 258
Cesaro summability (means) of Fourier series      12 14 246 250
Cesaro summability (means) of Hermite series      250 251
Cesaro summability (means) of Jacobi series      vi 246 248 249 256
Cesaro summability of Laguerre series      247 248 250 251 271
Cesaro summability of Legendre series      14 249
Cesaro summability of power series      325
Cesaro summability of ultraspherical series      248
Characteristic values      49 187
Charlier      see Polynomials
Christoffe, formula of      29 ff. 397
Christoffel      iii 42 43 47 402
Christoffel ( — Darboux) formula      42 ff. 318 326
Christoffel numbers      46 ff. 114 115 187 351 378
Clerc      see Bonomi — Clerc
Closure      see Orthogonal polynomials
Coifman — G. Weiss      273 417
Completely monotonic sequence      136 ff. 155
Confluent hypergeometric function      104
Conformal mapping      21 160 364 365 372
Conjugate function      279
Conjugate points (interpolation)      332 347
Continued fractions      v 54
Continuous operation      12
Convergence in mean      330
Convergence in mean, generalized      330
Convergence of interpolation      330
Convergence, quadrature      330 333 350
Convergent (continued fraction)      55 ff
Cotes numbers      12 349 363
Courant — Hilbert      see Hilbert — Courant
Cramer      251 402
Csordas — Williamson      388 418
Darboux      42 43 195 196 211 402
Darboux method      202 206 265 269 395
Darboux's formula for Jacobi polynomials      168 196 225 236 237 248 253 337 353
Davis — Hirschman      99 418
Davis — Rabinowitz      158 418
de la Vallee-Poisson summability      273
DeBruijn      110 418
Deviation, quadratic      38 41 288
Deviation, Tchebichef      41 368
Differences      34 35 136
Differential equation      v 16 37 152 159 164 166 210
Differential equation of Bessel functions      15
Differential equation of Hermite polynomials      106 176 380
Differential equation of Jacobi'(hypergeometric) polynomials      iv 60 117 141
Differential equation of Laguerre polynomials      100 117 176
Differential equation of ultraspherical polynomials      80 81
Dirichlet      85 402
Dirichlet — Mehler      see Integral representation
Dirichlet's integral      12 14
Discriminant of classical polynomials      143 ff
distribution      8 9 26 29 38 40 57 181 274 330 351 378
Distribution of Stieltjes type      9 33 38
Doetsch      34 35 380 402 418
Du Bois, Reymond      14 402
Eagleson      37 418
Egervary — Turan      348 418
ellipse      8 21 251
Ellipse of convergence      245 248 252 311 312
Elliptic functions, v      60
Equiconvergence      31 244 246 247 249 313
Erdelyi      ix 104 243 273 380 402 418
Erdelyi — Swanson      243 418
Erdos      vii 347 348 418
Erdos — Feldheim      334 402
Erdos — Grunwald      347 418
Erdos — Lengyel      347 418
Erdos — Turan      113 114 115 332 333 335 347 348 402 418
Euler      72 402
Euler's constant      15
Euler's integral of second kind      see Gamma function
Euler's integral of the first kind      14
expansion      v vi 58
Expansion in power series      313 314
Expansion in series of classical polynomials      244 ff. 251
Expansion in series of general orthogonal Hermite polynomials      244 245 247 251 253 269
Expansion in series of general orthogonal polynomials      313 ff
Expansion in series of Jacobi polynomials      244 ff
Expansion in series of Laguerre polynomials      244 ff. 249 253 266 313
Expansion in series of Legendre polynomials      248 249
Expansion in series of Tchebichef polynomials      248 314
Expansion in series of ultraspherical polynomials      248 249. Orthogonal
Expansion, finite cosine, of Legendre polynomials      90 ff
Expansion, infinite sine, of Legendre polynomials      90 ff
Faber      299 330 368 369 372 402.
Fatou      275 403
Fatou's theorem      274
Favard      43 403
Fejer      vii 14 45 58 89 91 96 134 136 157 165 172 174 175 179 198 202 249 264 330 336 348 350 386 403.
Fejer — Szego      175 404
Fejer's asymptotic formula for Laguerre polynomials      198 202 203 237 240 269
Fejer's generalization of Legendre polynomials      135 ff. 174 206
Fejer's integral      12
Fejer's representation of positive trigonometric polynomials      3 4 274
Fejer's representation of positive trigonometric polynomials, generalization of      275 ff
Fejer's second generalization of Legendre polynomials      137 138 155
Fekete      369 404
Feldheim      95 96 97 334 335 386 404 418.
Fitch      see Askey — Fitch
FOURIER      v
Fourier coefficient (constant)      24 287
Fourier series      12 14 24 25 28 38 39 244 246 253 274 289 311 314 323 347 367.
Fourier's inversion formula      387
Freud      309 418
Friedrichs      404
Fujiwara      145 404
Functions of second kind, Jacobi's      73 ff. 251
Functions of second kind, Legendre's      74 78 88 89 92 379 383. Integral Zeros
Fundamental polynomials of Hermite interpolation      330 ff
Fundamental polynomials of Lagrange interpolation      12 47 329
Galbrun      251 404
Gamma-function      14 75
Gasper      37 98 99 190 273 419.
Gatteschi      242 419
Gauss      iii 33 47 49 62 404. Mechanical
Gauss — Weierstrass summability      273
Gegenbauer      80 94 98 99 404
Generating function      35 36
Generating function of Hermite polynomials      106 380
Generating function of Jacobi polynomials      69 ff. 82 83 95
Generating function of Laguerre polynomials      101 202 379
Generating function of Legendre polynomials      90 206 207 383
Generating function of Pollaczek polynomials      393
Generating function of ultraspherical polynomials      82 83
Geometric mean      275 296 385
Geronimus      42 189 404
Gibbs' phenomenon      251
Ginibre      392 419
Gottlieb      37 405
Grenander — Szego      274 287 419
Griinbaum      392 419
Griinwald      vii 330 347 405 419.
Griinwald — Turan      346 405
Gronwall      165 249 405
Haar      14 248 405
Hahn, W.      33 107 111 112 129 132 151 405 419
Hamburger      57 110 405
hankel      see Quadratic form
Hardy      102 380 405
Hardy — Littlewood — Polya      2 405
Hartman — Wintner      20 419
Hausdorff      136 155 405
Heine      iii 27 37 47 54 80 90 151 192 194 251 405
Heine — Stieltjes theorem, iv      151 ff
Helly      13 405
Helly's theorem      13 14 258 272 330 339 350
hermite      v 156 405 406
Hermite interpolation      330 ff. 340 347 348
Hermite — Stieltjes      89 156 157 172 406
Hermitian form      287
hilb      195 249 406
hilbert      142 145 406
Hilbert — Courant      58 59 100 106 109 406
Hildebrandt      8 10 406
Hille      102 105 126 131 242 251 406.
Hirschman      99 273 419.
Hobson      iv 58 84 92 382 406
Hollo      335 406
Horton      99 273 419
Howell      392
Hsu      390 419
Hua      99 419
Hunt      273 420
Hurwitz, theorem of      21 150 193 239 371
Hypergeometric function, iii      62 ff. 83 84 96.
Inequalities      159 ff
Inequality for the arithmetic and geometric mean      2 300
Inequality, Abel's      2 174 205
Inequality, Cauchy's      2 39 120 160 183 291 302 306 321 370
Inequality, Schwarz's      2 9 110 136 162 268 276 317 376
Inequality, Turan's      190 388.
Integral equations      v 218 251
Integral representation of Legendre functions of second kind      88 ff
Integral representation of Legendre polynomials      85 ff
Integral representation of Legendre polynomials, Dirichlet — Mehler      85 ff. 96
Integral representation of Legendre polynomials, Laplace (first)      86 176
Integral representation of Legendre polynomials, Laplace (second)      86 ff
Integral representation of Legendre polynomials, Stieltjes      87 fl"
Integral representation of ultraspherical polynomials, Dirichlet — Mehler      89
Integral representation of ultraspherical polynomials, Stieltjes      89 ff
Integral, Lebesgue      vii 9 26 159 246 274 275 287 291 364 384
Integral, Riemann      9 10 280 281 282 298 310 333 335 351 361
Integral, Riemann — Stieltjes      8 9 11 50 333 350 351
Integral, Stieltjes — Lebesgue      v 1 8 10 38 187
Interpolation      v ix 12 14 58 329 347
Interpolation on Hermite abscissas      340
Interpolation on Jacobi abscissas      335 ff
Interpolation on Laguerre abscissas      340 344
Interpolation on Legendre abscissas      386
Interpolation on Tchebichef abscissas      330 334 335 386
Interpolation on ultraspherical abscissas      386. See Conjugate points Fundamental Hermite Lagrange
Interpolation, Lagrange      12 47 180 329 348 349 389
Jackson      vii 6 7 42 331 406
Jacob      251 406
Jacobi, iii      49 58 69 86 406 407. Mechanical Polynomials Series
Jensen's theorem      301
Joo      391
Jordan arc      8 364
Jordan curve      8 21 364 365
Jordan — Pochhammer integral      75
Jordan, C.      58 407
Jordan, Ch.      34 407
Julia      369 407
Kaczmarz — Steinhaus      1 10 407
Kanter      vii
Karlin — McGregor      37 273 389 420
Karlin — Studden      5 420
Karlin — Szego      190 388 420
Keldysch — Lawrentieff      368 407
Kernel polynomials      39 40 44 71 101 180 183 249 290 322 323 333 351 368 377. Zeros
Klein      145 407
Kogbetliantz      102 168 172 203 240 248 249 251 256 273 380 407
Koornwinder      98 99 420
Korous      131 132 162 167 212 251 303 408
Koschmieder      60 408
Kowalewski      24 408
Kowallik      251 408
Krall      107 408
Krawtchouk      113 408.
kronecker      408
l'Hospital, rule of      307
Lagrange      100 408
Lagrange polynomials      329 ff. 347
Lagrange series      70. See Interpolation
Laguerre      100 117 131 408 409.
Lame function      151
Langer      204 210 409
Laplace      iv 86
Laplace series      96 249
Laplace series, transform      379. See Integral representation
Laplace — Heine formula      194 204 208 243
Laplace — Heine formula, generalization      194
Laplace's formula for Legendre polynomials      194 198 201 202 204 211 224
Laplace's formula for Legendre polynomials, Darboux's generalization      195 201 206 211
Laplace's formula for Legendre polynomials, Stieltjes' generalization      195 202 209
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