By J.C. Mason

Chebyshev polynomials crop up in nearly each quarter of numerical research, and so they carry specific significance in contemporary advances in topics akin to orthogonal polynomials, polynomial approximation, numerical integration, and spectral equipment. but no booklet devoted to Chebyshev polynomials has been released considering that 1990, or even that paintings centred totally on the theoretical facets. A extensive, up to date therapy is lengthy overdue.Providing hugely readable exposition at the subject's cutting-edge, Chebyshev Polynomials is simply the sort of remedy. It contains rigorous but down-to-earth insurance of the speculation besides an in-depth examine the homes of all 4 sorts of Chebyshev polynomials-properties that result in a variety of ends up in components comparable to approximation, sequence expansions, interpolation, quadrature, and vital equations. difficulties in every one bankruptcy, ranging in trouble from ordinary to particularly complex, strengthen the techniques and techniques presented.Far from being an esoteric topic, Chebyshev polynomials lead one on a trip via all parts of numerical research. This ebook is the fitting car with which to start this trip and one who also will function a customary reference for a few years to come back.

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C[a, b]: continuous functions on [a, b]; 2. L∞ [a, b]: bounded functions on [a, b]; 3. L2 [a, b]: square-integrable functions on [a, b]; 4. 1) where w(x) is a given non-negative weight function and 1 ≤ p < ∞. Note that L2 [a, b] is a special case (p = 2) of Lp [a, b]. © 2003 by CRC Press LLC The reason for deﬁning such a family of functions, when in practice we may only in fact be interested in one speciﬁc function, is that this helps to isolate those properties of the function that are relevant to the theory — moreover, there is a close link between the function class we work in and the norms we can use.

Pn−1 (x) −2x 1 Pn (x) 1 −2x 1 1 X 0 0 .. 22a) or (denoting the (n + 1) × (n + 1) matrix by A) Ap = c where c= 1 X 0 .. 22b) 0 with X = −x, 0, −1, 1, respectively in the four cases. Let bT = (b0 , b1 , . . , bn ) © 2003 by CRC Press LLC be the row vector satisfying the equation 1 −2x 1 1 −2x 1 1 −2x 1 (b0 , b1 , . . , bn ) = .. .. . . 1 −2x 1 1 −2x 1 = (a0 , a1 , . . 23a) or bT A = aT .

On transforming to x = cos θ, this leads to the required formula for Tn (x) in terms of xn , xn−2 , xn−4 , . . © 2003 by CRC Press LLC We omit the details here, but refer to Rivlin (1974), where the relevant result is obtained in the form n/2 n/2 n j n−2k (−1)k x . 16) where (n) ck = (−1)k 2n−2k−1 2 n−k n−k−1 − k k and (2k) ck = (−1)k (2k < n) (k ≥ 0). 3a), and we leave this as an exercise for the reader (Problem 5). 17a) may be further simpliﬁed, by taking out common ratios, to give (n) ck = (−1)k 2n−2k−1 n−k n .