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| Snyder M.A. — Chebyshev methods in numerical approximation |
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| Предметный указатель |
Addition formula 26 98
Approximation, reasons for 1 2
Arctangent, Chebyshev series for 35 51 52
Bessel functions 45 46 47 56 93—96
Bessel functions, Chebyshev expansions of 49
Bessel functions, differential recurrence relation 20
Bessel functions, Fourier transform of 25
Bessel functions, relation to Chebyshev polynomials 19 20 35 47—49 101
Chebyshev differential equation 12 13 23 98
Chebyshev expansions of absolute value function 56
Chebyshev expansions of algebraic powers 45
Chebyshev expansions of arctangent 35 51 52
Chebyshev expansions of Bessel functions 47—49
Chebyshev expansions of cosine 47
Chebyshev expansions of delta distribution 56
Chebyshev expansions of exponential 45 49
Chebyshev expansions of hyperbolic cosine 47
Chebyshev expansions of hyperbolic sine 46
Chebyshev expansions of inverse monomials 49 50
Chebyshev expansions of logarithm 50 52
Chebyshev expansions of sign function 56
Chebyshev expansions of sine 46
Chebyshev expansions, coefficients of 29 100 102
Chebyshev functions 13
Chebyshev polynomials and analytic functions 30 101
Chebyshev polynomials and Bessel functions 19 20 47— 49 101
Chebyshev polynomials and continued fractions 78 80 81 82
Chebyshev polynomials and difference equations 57
Chebyshev polynomials and economization of continued fractions 82—86
Chebyshev polynomials and economization of power series 36—39
Chebyshev polynomials and hypergeometric functions 25 98
Chebyshev polynomials and matrices 26
Chebyshev polynomials and optimal approximation 55
Chebyshev polynomials and powers 36 39 40 45 100 102 104 105
Chebyshev polynomials and trigonometric functions 9 13 45—47 97
Chebyshev polynomials and ultraspherical polynomials 8 32 33
Chebyshev polynomials of the first kind 8 9 11 19 39 102
Chebyshev polynomials of the second kind 21 22 23 102 103
Chebyshev polynomials, addition formula 26 98
Chebyshev polynomials, and hyperbolic functions 46 47 111
Chebyshev polynomials, behavior near 1 18
Chebyshev polynomials, completeness 27—28
Chebyshev polynomials, derivatives of 22 23 101
Chebyshev polynomials, expansions in 29 30 32 33 44—56 100 102
Chebyshev polynomials, explicit representations 9 14 18 21 23 24 25 99 102 103 104
Chebyshev polynomials, functional equations 15 26 98
Chebyshev polynomials, generating functions 15—16 23 26 44 99
Chebyshev polynomials, integrals of 15 20 23 63 100 102
Chebyshev polynomials, multiplication formulas 15 99
Chebyshev polynomials, orthogonality of 16 20 29 98 101
Chebyshev polynomials, partial fraction decomposition 25 101
Chebyshev polynomials, products of powers with 15 21 40 102
Chebyshev polynomials, rational expressions in 10 80 81
Chebyshev polynomials, recurrence relations 9 15 22 25 97 98
Chebyshev polynomials, Rodrigue’s formula 16 47 101
Chebyshev polynomials, shifted, of the first kind 20 21 40 62 63 101 103
Chebyshev polynomials, shifted, of the second kind 22 24 101 104
Chebyshev polynomials, summation of series in 34 35 100 101 102
Chebyshev polynomials, summation orthogonality 30
Chebyshev polynomials, zeros of 14 25 101
Chebyshev’s theorem 31 32
Completeness 27—28
Continued fractions 9 10 78
Continued fractions and differential equations 76—78
Continued fractions, accelerated convergence of 72 73
Continued fractions, classic expansions 69—72
Continued fractions, convergence of 68 69
Continued fractions, conversion to 66 81
Continued fractions, evaluation on computer 10
Continued fractions, Thiele expansions 73—76
Contour integrals 12 48 93 94
Convergence of continued fractions 68 69
Delta distribution 56 57
Delta, Kronecker 6 16
Derivative of Chebyshev polynomials 22 23 101
Derivative, inverse 73
Derivative, reciprocal 73
Difference equations 41 50 57 91 92
Difference, reciprocal 73
Difference, symmetric 73
Differential equations and continued fractions 76—78
Differential equations and tau method 42—43
Differential equations, Chebyshev 12—13 14 23 98
Dirac delta function 56 57
| Economization of continued fractions 82—86
Economization of power series 36—39
Error curve, standard form 87
Expansions (see Chebyshev expansions)
Exponential function 1 45
Exponential function, Chebyshev series for 45 49
Exponential function, economized series for 4 37 38 39
Exponential integral function 42—44
Functional equations 15 26 98
Gauss duplication formula 48
Generating functions 15—16 23 26 44 99
Hyperbolic functions 46 47
Hypergeometric function 25 98
Integrals of Chebyshev polynomials 15 20 23 63 100 102
Inverse tangent (see Arctangent)
Kopal’s method 61—63
Kronecker delta 6 16
Lanczos (see Tau method)
Least squares norm 3 6
Least squares norm of Legendre polynomials 7
Legendre polynomials 6 7
Legendre polynomials, expansions in 7 34
Logarithm, Chebyshev series for 50 52
Logarithm, rational approximations to 60 62 63 65 66
Maehly’s method 80—82
Matrix and Chebyshev polynomials 26
Maximum norm 3 4
minimax (see Optimal approximations)
Multiplication formulas 15 99
Norms 3
ObreschkofFs formula 63—66
ObreschkofFs formula, optimal 86
Optimal approximations, form of error 4
Optimal approximations, polynomials 53—55
Optimal approximations, rational 86—89
Orthogonal polynomials 5 6 7 34
Orthogonality of Chebyshev polynomials 16 20 29 98 101
Orthogonality of Legendre polynomials 6
Orthogonality of ultraspherical polynomials 8 32
Orthogonality, under summation 30
pade 58—60
Pade approximation 58—60
Pade of power function 65
Pade of square root function 61
Pade table 59
Partial fractions 25 101
Polynomials, Chebyshev (see Chebyshev polynomials)
Polynomials, evaluation of 2
Polynomials, Legendre 6 7 34
Polynomials, minimax 53
Polynomials, optimal 53
Polynomials, orthogonal 5 6 7 34
Polynomials, ultraspherical 8 32 33
Polynomials, zeros of 7 14 25 101
Powers, as combinations of Chebyshev polynomials 36 39 40 45 100 104 105
Products of powers and Chebyshev polynomials 15 21 40 102
Rational approximation by iteration 60 61
Rational approximation of exponential function 65
Rational approximation of logarithm function 60 62 63 65 66
Rational approximation with Chebyshev polynomials 10 80 81
Rational approximation, advantages of 9
Rational approximation, conversion to continued fractions 66—68
Rational approximation, Kopal’s method 61—63
Rational approximation, minimax 86
Recurrence relations for Chebyshev polynomials 9 15 22 25 97 98
Recurrence relations for Legendre polynomials 6
Recurrence relations for modified Bessel functions 20
Ricatti equation 76
Rodigue’s formula for Chebyshev polynomials 16 47 101
Rodigue’s formula for Legendre polynomials 6
Rodigue’s formula for ultraspherical polynomials 8 32
Shifted polynomials (see Chebyshev polynomials)
Sign function 56
Summation of series 34 100 101 102
Summation orthogonality 30
Tau method 40—44 49 61
Taylor’s series 1 5 64
Taylor’s series, inadequacy of 4 5 34 54
Taylor’s series, relation to ultraspherical polynomials 8 34
Thiele expansions 73—76
Trigonometric functions 9 13 46 47 97
Ultraspherical polynomials 8 32 33
Zeros of Chebyshev polynomials 14 25 101
Zeros of orthogonal polynomials 7
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