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20 Polynomial Approximation of Differential Equations Figure 1.7.3 - Scaled Hermite functions for 1 ≤ n ≤ 9. * * * * * * * * * * * * Our survey is now concluded. Of course, other families of polynomials could be consid- ered. However, the ones presented here are those actually used in spectral methods.
2 ORTHOGONALITY Inner products and orthogonal functions are fundamental concepts in approximation theory. All the families introduced in the first chapter form an orthogonal set of functions with respect to a suitable inner product. This property will reveal many other interesting features. 2.1 Inner products and norms In the following, X will denote a real vector space, i.e. a set in which addition and scalar multiplication are defined with the usual basic properties (for the unfamiliar reader a wide bibliography is available; we suggest, for instance, hoffman and kunze(1971), lowenthal(1975)). An inner product (· , ·) : X × X → R is a bilinear application. This means that, for any couple of vectors in X, their inner product is a real number and the application is linear for each one of the two variables. Moreover, we require symmetry, i.e. (2.1.1) (u,v) = (v,u), ∀u,v ∈ X, and the application has to be positive-definite, i.e. (here 0 is the zero of X): (2.1.2) (u,u) ≥ 0, ∀u ∈ X, (u,u) = 0 ⇐⇒ u ≡ 0.
Page 7 and 8: PREFACE This book is devoted to the
Page 9 and 10: CONTENTS 1. Special families of pol
Page 11 and 12: Contents ix 7. Derivative Matrices
Page 13 and 14: 1 SPECIAL FAMILIES OF POLYNOMIALS A
Page 15 and 16: Special Families of Polynomials 3 n
Page 17 and 18: Special Families of Polynomials 5 B
Page 19 and 20: Special Families of Polynomials 7 1
Page 21 and 22: Special Families of Polynomials 9 (
Page 23 and 24: Special Families of Polynomials 11
Page 31: Special Families of Polynomials 19
Page 35 and 36: Orthogonality 23 The function w is
Page 37 and 38: Orthogonality 25 As we promised, we
Page 39 and 40: Orthogonality 27 Let us now define
Page 41 and 42: Orthogonality 29 Therefore, one obt
Page 43 and 44: Orthogonality 31 From (2.3.8), we g
Page 45 and 46: Orthogonality 33 only mention the r
Page 47 and 48: 3 NUMERICAL INTEGRATION As we shall
Page 49 and 50: Numerical Integration 37 In the cas
Page 51 and 52: Numerical Integration 39 (3.1.9) z
Page 53 and 54: Numerical Integration 41 We give th
Page 55 and 56: Numerical Integration 43 where 1
Page 57 and 58: Numerical Integration 45 (3.2.10)
Page 59 and 60: Numerical Integration 47 This means
Page 61 and 62: Numerical Integration 49 (3.4.2) w
Page 63 and 64: Numerical Integration 51 3.5 Gauss-
Page 65 and 66: Numerical Integration 53 The proof
Page 67 and 68: Numerical Integration 55 3.7 Clensh
Page 69 and 70: Numerical Integration 57 the first
Page 71 and 72: Numerical Integration 59 (3.8.14) p
Page 73 and 74: Numerical Integration 61 (3.9.5) p
Page 75: Numerical Integration 63 (3.10.5)
Page 78 and 79: 66 Polynomial Approximation of Diff
Page 80 and 81: 68 Polynomial Approximation of Diff
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70 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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