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| Jackson D. — Fourier Series and Orthogonal Polynomials |
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| Ïðåäìåòíûé óêàçàòåëü |
Asymptotic formula for  87—88
Bessel functions of order  217
Bessel functions of order n 80—88 217—19
Bessel functions of order zero 69—80 216 219
Bessel series 79—80 83—84 109—114 138—141 216
Birkhoff, G.D. 90 229
Bocher, M. 24 229
Boundary value problems 88—90 91—141 220—223
Bounds of Legendre polynomials 61—63
Bounds of normalized Jacobi polynomials 191—92 200—201
Bounds of other orthonormal polynomials 201—208
Broken-line function 13—14 21
Byerly, W.E. 112 220
Carslaw, H.S. 229
Charlier, C 178 179
Christoffel — Darboux identity 157—58
Christoffel, E.B. 54 55 157 158 206
Christoffel’s identity 54—55
Churchill, R.V. 229
Convergence of Fourier series 12—14 18—25 42—44
Convergence of of Legendre series 63—68 215
Convergence of of more general series of orthogonal polynomials 191—208
Cosine series 6—7 8—9
Cosine sum 36
Courant, R. 88 189 229
Cylindrical coordinates 106 109—114 138—141
Damped vibrations 100—101 221
Darboux, G. 157 158 175 206 229
de la Valine Poussin, C.J. 35
de Moivre, A. 126
Derivative formula for Hermite polynomials 176—77
Derivative formula for Jacobi polynomials 166—167
Derivative formula for Laguerre polynomials 184
Derivative formula for Legendre polynomials 57—58
Differential equation for  69—71 74—76 89 219
Differential equation for  80—82
Differential equation for a class of orthogonal polynomials 161—165
Differential equation for derivatives of Legendre polynomials 119—120
Differential equation for Hermite polynomials 180 226
Differential equation for Jacobi polynomials 173—174 225
Differential equation for Laguerre polynomials 186 226—227
Differential equation for Legendre polynomials 48—49 55—57 88—89
Differential equation for Pearson frequency functions 142—147
Differential equation for sines and cosines 88
Differential equation for Sturm — Liouville functions 89—90
Discontinuities 9—11 22—24 65—67
Double Fourier series 115—117
Elderton, W.Palin 229
Even function 6—7
Fejer, L. 32 229
Fejer’s theorem 32—35
Finite jump 19
Flow of heat 92
for polynomial approximation 212
Fourier series 1—44 91—101 103—105 111—112 115—117 209—211
Frank, P. 230
Frequency functions 142—148 223
Function (general notion) 10
Gauss, K.F. 55
Generating function for Hermite polynomials 181
Generating function for Jacobi polynomials 174—75
Generating function for Laguerre polynomials 187—188
Generating function for Legendre polynomials 45—46
Generating function for ultraspherical polynomials 225—26
Gibbs phenomenon 24
Gibbs, T.Willard 24
Gram — Charlier series 179
Gram, J.P. 178 179
Gray, A. 229
Green, G. 103 107
Harmonic polynomials 126—132 137—138
Heine, E. 175
Hermite polynomials 176—183 226
Hermite series 178—179
Hilbert, D. 88 189 229
Hille, E. 230
Hobson, E.W. 229
Ince, E.L. 90 229
Integrability 5
Integrability in the sense of Lebesgue 5 197—99
Integral representation of 74—76
Integral representation of 84—85 218
Integral representation of  58—60
Jackson, D. 44
Jacobi polynomials 166—175 200—201 225—226
Jacobi series 172 201
Kaczmarz, S. 229
Kellogg, O.D. 229
Korous, J. 205
Laguerre polynomials 184—190 226—227
Laguerre series 186
Laplace series 121—138 222
Laplace’s equation 91—96 101—112 115—141 220—222
Leading coefficients of Hermite polynomials 177
Leading coefficients of Jacobi polynomials 169—171 172
Leading coefficients of Laguerre polynomials 184
Leading coefficients of Legendre polynomials 47—48
Least-square property of Fourier series 27—29
Least-square property of general orthogonal polynomials 160—161
Least-square property of Legendre series 215—216
Lebesgue constants 40—42
| Lebesgue, H. 5 40 41 42 43 44 196 197 229
Legendre polynomials 45—68 213—216
Legendre series 53 63—68 107—209 214—16
Leibniz, G.W. 166 167 169 184 186 187
Liouville, J. 89
Lipschitz, R. 44
Mac Lane, S. viii
Maclaurin, C. 49
MacRobert, T.M. 229
Marden, M. 162
Mathews, G.B. 229
Moments 144—48
Moore, C.N. 84
Normalizing factor for Bessel functions 73—74 83
Normalizing factor for derivatives of Legendre polynomials 124—125
Normalizing factor for Hermite polynomials 177—178
Normalizing factor for Jacobi polynomials 171—172
Normalizing factor for Laguerre polynomials 185
Normalizing factor for Legendre polynomials 51—52
Odd function 7
Orthogonal polynomials 45—68 123—125 149—208 213—216 223—228
Orthogonality 3 224
Orthogonality, of Bessel functions 71—73 82—84
Orthogonality, of functions of two variables 117 122—124 140
Orthogonality, of Hermite polynomials 177 226
Orthogonality, of Jacobi polynomials 167—168
Orthogonality, of Laguerre polynomials 184—185 227
Orthogonality, of Legendre polynomials 50—51
Orthogonality, of sines and cosines 2—3
Orthonormal functions 152
Orthonormal polynomials 154
Parseval, M.A. 29 31 216
Parseval’s theorem 29—31
Pearson, K. 142 147 162 165 230
Peebles, G. 203
Period 1
Po1ya, G. 230
Poisson, S.D. 103 105 136 137
Poisson’s integral 105 136—37
Polar coordinates 101—103
Probability integral 178
Recurrence formulas for Bessel functions 85—87 218
Recurrence formulas for general orthogonal polynomials 156—157
Recurrence formulas for Hermite polynomials 179—180
Recurrence formulas for Jacobi polynomials 172—173
Recurrence formulas for Laguerre polynomials 186—187
Recurrence formulas for Legendre polynomials 46—47 58—60
Recurrence formulas for sines and cosines 211—212 225
Riemann — Weber 230
Riemann, B. 14 16 21 25
Riemann’s theorem 14—16
Rietz, H.L. 230
Robertson, H.P. 230
Rodrigues, O. 57 166 167
Rodrigues’s formula 57—58
Rolle, M. 78
Schmidt, E. 151 152 224 225
Schmidt’s process of orthogonalization 151—153
Schrbdinger equation 181—183 188—190
Schrodinger, E. 182 188
Schwarz, H.A. 38 39 195 202 204 207 215 224 228
Schwarz’s inequality 38 224
Shohat, J. 230
Sine series 7 8—10 94 95 98—99 100 101 112
Singular points of differential equations 70 88—89
Spherical coordinates 106—109 118—120 189
Spherical harmonics 118—138 189
Steinhaus, H. 229
Stone, M.H. 230
Sturm — Liouville boundary value problem 89—90
Sturm, C. 89
Summation of series 31—35
Szego, G. 174 205 230
Taylor, B. 78
Tchebichef, P.L. 150 230
Titchmarsh, E.C 230
Tonelli, L. 230
Transformation of Laplace’s equation 101—103 105—107 129—131
Trigonometric sum 25—26
Ultraspherical polynomials 225—226
Uniform convergence 21—22 42—44 215
Uspensky, J.V. 230
Van der Waerden, B.L. 189 230
Vibrating membrane 112—114 221 222—223
Vibrating string 96—101 220—221
von Mises, R. 230
Walsh, J.L. 230
Watson, G.N. 230
Wave equation of hydrogen atom 188—190
Wave equation of linear oscillator 181—183
Weierstrass, K. 25 26 32 35 40 137 212 216 227
Weierstrass’s theorem for trigonometric approximation 25—27 35—40
Weight function 72 149—150 153—154
Weight function for Hermite polynomials 176
Weight function for Jacobi polynomials 166
Weight function for Laguerre polynomials 184
Weyl, H. 183 189 230
Whittaker, E.T. 230
Zeros of Bessel functions 72—73 76—79 82 87 218 219
Zeros of general orthogonal polynomials 159—160
Zeros of Legendre polynomials 52—53
Zygmund, A. 24 230
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