| Preface |
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ix | |
| Preface to the English Edition |
|
xv | |
| Notation |
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xix | |
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General properties of polynomials orthogonal over a domain |
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1 | (36) |
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Polynomials in two variables orthogonal over a domain |
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1 | (5) |
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The existence theorem and criteria of orthogonality |
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6 | (4) |
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10 | (8) |
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Monic orthogonal polynomials |
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18 | (6) |
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Normal biorthogonal systems |
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24 | (4) |
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Fourier series of orthogonal polynomials in two variables |
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28 | (3) |
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Fourier series for differentiable functions |
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31 | (6) |
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Some typical examples and special cases of orthogonality over a domain |
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37 | (26) |
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Different products of classical orthogonal polynomials |
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37 | (5) |
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Various cases of connection between orthogonality over a domain and orthogonality on an interval |
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42 | (6) |
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Some theorems in the case of a weight function with separating variables |
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48 | (4) |
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Conditions of interconnection between the weight function and the domain of orthogonality |
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52 | (5) |
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Examples of computations of weight function moments |
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57 | (6) |
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Classical Appell's orthogonal polynomials |
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63 | (24) |
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Rodrigues formula for Appell's polynomials |
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63 | (6) |
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Representation of the Appell polynomials via the hypergeometric function of two variables |
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69 | (3) |
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Differential equation for the Appell polynomials |
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72 | (3) |
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Orthogonality of eigenfunctions of the Appell equation |
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75 | (4) |
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Normal biorthogonal Appell system |
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79 | (4) |
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Series in the Appell polynomials |
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83 | (4) |
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Admissible differential equation for polynomials orthogonal over a domain |
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87 | (44) |
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The main differential operator and a theorem on orthogonality |
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87 | (5) |
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Admissibility conditions for the main differential equation |
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92 | (5) |
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Some examples and properties of admissible differential equations |
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97 | (4) |
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Affine transformations of the arguments of the main differential equation |
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101 | (4) |
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Transformation of the coefficients of the characteristic polynomial |
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105 | (10) |
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Normal forms of the admissible differential equation |
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115 | (8) |
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Normal forms when reducing the degree of the characteristic polynomial |
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123 | (8) |
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Potentially self-adjoint equation and Rodrigues formula |
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131 | (32) |
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Potentially self-adjoint operators |
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131 | (4) |
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Admissible and potentially self-adjoint equations |
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135 | (11) |
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Rodrigues formula for polynomials orthogonal over a domain |
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146 | (7) |
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Weight functions and the Rodrigues formula in the most typical cases |
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153 | (10) |
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Harmonic polynomials orthogonal over a domain |
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163 | (24) |
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Homogeneous harmonic polynomials |
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163 | (6) |
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An analogue of the Christoffel-Darboux formula |
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169 | (4) |
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Harmonic polynomials orthogonal in the unit disk |
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173 | (3) |
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Harmonic polynomials orthogonal over a domain in the general case |
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176 | (3) |
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Harmonic polynomials superorthogonal over a domain |
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179 | (8) |
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Polynomials in two variables orthogonal on a contour |
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187 | (36) |
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Main definitions and the simplest properties |
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187 | (4) |
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Polynomials in two variables orthogonal on an algebraic curve |
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191 | (5) |
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Harmonic polynomials orthogonal on a contour |
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196 | (4) |
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Fourier series in harmonic polynomials orthogonal on a contour |
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200 | (6) |
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Harmonic polynomials superorthogonal on a contour |
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206 | (7) |
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Examples of superorthogonal systems of harmonic polynomials |
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213 | (10) |
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Generalized orthogonal polynomials in two variables |
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223 | (30) |
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Main definitions and the simplest properties |
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223 | (5) |
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The existence theorem in the most general form |
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228 | (5) |
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Fourier series in generalized orthogonal polynomials in two variables |
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233 | (8) |
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Monic orthogonal polynomials under minimal conditions |
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241 | (6) |
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Generalized generating functions for monic orthogonal polynomials |
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247 | (6) |
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Other results concerning the connection between orthogonal polynomials and differential equations |
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253 | (32) |
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Canonical admissible differential equation and monic orthogonal polynomials |
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253 | (5) |
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Necessary consistency conditions of the canonical operator and the functional |
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258 | (4) |
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Sufficient conditions of consistency of the canonical operator and the functional |
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262 | (6) |
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The deduction of the differential equation from the Pearson equation system |
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268 | (8) |
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An admissible partial differential equation of an arbitrary order |
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276 | (9) |
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Original results of T. Koornwinder |
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285 | (28) |
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Examples of the representation of polynomials orthogonal over a domain via the Jacobi polynomials |
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285 | (6) |
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Orthogonal polynomials in tow conjugate complex variables |
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291 | (5) |
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The Chebyshev polynomials in two conjugate complex variables for the Steiner domain |
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296 | (12) |
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Another generalization of the Jacobi polynomials onto the case of two variables |
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308 | (5) |
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313 | (10) |
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A new generalization of the Appell polynomials |
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313 | (5) |
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Some properties of the Koornwinder-Steiner polynomials |
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318 | (1) |
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A two-dimensional analogue of the Chebyshev-Laguerre polynomials |
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319 | (4) |
| Comments and Supplements |
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323 | (6) |
| References |
|
329 | (14) |
| Author Index |
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343 | (2) |
| Subject Index |
|
345 | |
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