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68 Polynomial Approximation of Differential Equations Henceforth, depending on the basis used for the representation of a polynomial, we distinguish among the two cases with the help of the following terminology. When the space of polynomials is intended to be the span of the uk’s, then it is isomorph to the set of Fourier coefficients. In this situation we call it the frequency space. Otherwise, when the isomorphism is naturally established with the set of the point values, it will be denoted by physical space. There are situations in which it is more convenient to work with one of the two representations. The discrete Fourier transform allows passage from the physical to the frequency space, thus both representations can be used to their fullest advantage. The cost of implementing the transform or its inverse is proportional to n 2 , since it corresponds to a matrix-vector multiplication. Most of this chapter is dedicated to the analysis of a special case which occurs when Chebyshev polynomials are used. For this choice of polynomials, fast algorithms have been developed to perform the discrete Fourier transform, considerably reducing the computational expense. 4.2 Aliasing Two operators Πw,n and Iw,n have been respectively introduced in sections 2.4 and 3.3. For a continuous function f, both the images Πw,n−1f and Iw,nf are in Pn−1, which is interpreted as frequency or physical space respectively. Unless f is a polynomial in Pn−1, the projection and the interpolant do not coincide. We show two examples. In figures 4.2.1 and 4.2.2, we plot in [−1,1] the projection and the interpolant of the same degree, corresponding to two different functions. In the first figure we have set f(x) = 4|x| − 2 − 1, w(x) ≡ 1 (Legendre weight function) and n = 8. In the second we have set f(x) = 1 2 − |sinπx|, w(x) = 1/ √ 1 − x 2 (Chebyshev weight function) and n = 13.
Transforms 69 Figure 4.2.1 - Legendre projection and interpolant of degree 7 for the function f(x) = 4|x| − 2 − 1, x ∈ [−1,1]. Figure 4.2.2 - Chebyshev projection and interpolant of degree 12 for the function f(x) = 1 2 − |sin πx|, x ∈ [−1,1].
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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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Page 33 and 34: 2 ORTHOGONALITY Inner products and
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 112 and 113: 100 Polynomial Approximation of Dif
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118 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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