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152 Polynomial Approximation of Differential Equations we are concerned with finding n + 1 complex polynomials pn,m, 0 ≤ m ≤ n, of degree at most n, and n + 1 complex numbers λn,m, 0 ≤ m ≤ n, such that (8.1.1) ⎧ ⎨ ⎩ p ′ n,m(η (n) i ) = λn,m pn,m(η (n) i ), 1 ≤ i ≤ n, pn,m(η (n) 0 ) = λn,m pn,m(η (n) 0 ), 0 ≤ m ≤ n. Each pn,m, 0 ≤ m ≤ n (different from the zero function) is determined up to a constant factor. Let us suppose that pn,m(η (n) 0 ) = 0, for m = 0. Then, it is easy to realize that λn,0 = 1. On the contrary, the remaining λn,m, 1 ≤ m ≤ n, are eigenvalues of the n × n matrix, obtained by assuming σ = 0 in (7.4.1) (in (7.4.3) we have an example for n = 3). Therefore, except for m = 0, the eigenfunctions satisfy pn,m(η (n) 0 ) = 0. In figures 8.1.1 to 8.1.4, we give the plot of the eigenvalues λn,m, 1 ≤ m ≤ n, in the window [−4,12] × [−12,12] of the complex plane. We considered respectively the four cases: α = β = −.7, −.5, 0, .5 . The integer n varies from 4 to 10. The eigenvalues, corresponding to the same value of n, have been joined together by segments. In the first three cases, all the eigenvalues lie in the half plane of the complex numbers with positive real part. In general, we are induced to conjecture that Reλn,m > 0, 1 ≤ m ≤ n, when α > −1, −1 < β ≤ 0. Unfortunately, we do not have the proof of this fact. The statement has been proven in solomonoff and turkel(1989), for the Chebyshev case (α = β = −.5), and could be adapted to other situations. Anyway, the proof is quite technical and we omit it. We further characterize the eigenvalues with the following proposition. Theorem 8.1.1 - For 1 ≤ m ≤ n, λn,m satisfies the relation (8.1.2) λ n n,m + n j d j=1 dx j ˜ l (n) 0 (−1) where the Lagrange polynomial ˜ l (n) 0 is defined in (3.2.8). λ n−j n,m = 0,
Eigenvalue Analysis 153 Figure 8.1.1 - Eigenvalues {λn,m}1≤m≤n Figure 8.1.2 - Eigenvalues {λn,m}1≤m≤n for 4 ≤ n ≤ 10 and α = β = −.7 . for 4 ≤ n ≤ 10 and α = β = −.5 . Figure 8.1.3 - Eigenvalues {λn,m}1≤m≤n Figure 8.1.4 - Eigenvalues {λn,m}1≤m≤n for 4 ≤ n ≤ 10 and α = β = 0 . for 4 ≤ n ≤ 10 and α = β = .5 .
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PREFACE This book is devoted to the
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CONTENTS 1. Special families of pol
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Contents ix 7. Derivative Matrices
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1 SPECIAL FAMILIES OF POLYNOMIALS A
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Special Families of Polynomials 3 n
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Special Families of Polynomials 5 B
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Special Families of Polynomials 7 1
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Special Families of Polynomials 9 (
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Special Families of Polynomials 11
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2 ORTHOGONALITY Inner products and
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Orthogonality 23 The function w is
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Orthogonality 25 As we promised, we
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Orthogonality 27 Let us now define
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Orthogonality 29 Therefore, one obt
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Orthogonality 31 From (2.3.8), we g
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Orthogonality 33 only mention the r
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3 NUMERICAL INTEGRATION As we shall
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Numerical Integration 37 In the cas
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Numerical Integration 39 (3.1.9) z
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Numerical Integration 41 We give th
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Numerical Integration 43 where 1
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Numerical Integration 45 (3.2.10)
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Numerical Integration 47 This means
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Numerical Integration 49 (3.4.2) w
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Numerical Integration 51 3.5 Gauss-
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Numerical Integration 53 The proof
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Numerical Integration 55 3.7 Clensh
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Numerical Integration 57 the first
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Numerical Integration 59 (3.8.14) p
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Numerical Integration 61 (3.9.5) p
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Numerical Integration 63 (3.10.5)
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66 Polynomial Approximation of Diff
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100 Polynomial Approximation of Dif
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REFERENCES adams r.(1975), Sobolev
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References 295 canuto c., hussaini
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References 297 friedman a.(1959), F
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References 299 jackson d.(1930), Th
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References 301 pavoni d.(1988), Sin
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References 303 vainikko g.m.(1964),
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