New PDF release: Analysis and partial differential equations. Dedicated to

By Cora Sadosky

ISBN-10: 0824783026

ISBN-13: 9780824783020

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Extra info for Analysis and partial differential equations. Dedicated to Mischa Cotlar

Example text

Sin xn sin 2xn sin nxn n cos xj − cos xk . 17) k=1 and take 2n equidistant points ξk := (2k − 1)π 2n (k = 1, . . , 2n). 17) can be expressed in the form tn (x) = an cos nx + 2n cos nx 2n (−1)k cot k=1 x − ξk tn (ξk ). 18) Proof Since ξk (k = 1, . . 19) where A is a constant. Notice also that for an arbitrary k ∈ {1, . . , 2n} x − ξk x − (π + ξk ) cos sin x − ξk 2 =− 2 cot = , x − ξk x − ξk 2 sin sin 2 2 so that we can conclude that Pk∗ (x) := x − ξk (−1)k cos nx cot 2n 2 is a trigonometric polynomial of degree n.

1) becomes the wellknown Vandermonde determinant 1 x1 · · · x1n−1 . = D = V (x1 , . . , xn ) = .. 1 xn xnn−1 n (xk − xj ). 4) j

268] and [431]). 1 Representations and Computation of Interpolation Polynomials In Sect. 2 we have already introduced a polynomial of minimal degree which interpolates a function f at n fixed points x1 , x2 , . . , xn . 4 Interpolation by Algebraic Polynomials 49 as the Lagrange interpolation polynomial and it was discovered in 1795 by Joseph Louis Lagrange. Otherwise, interpolation by polynomials is a very old subject in mathematics. The first formula with equally spaced points {xk } was found in the 1670’s by Isaac Newton.

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Analysis and partial differential equations. Dedicated to Mischa Cotlar by Cora Sadosky


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