Tchebycheff Polynomials and Approximation Theory Matthew A Melton
Tchebycheff Polynomials and Approximation Theory Matthew A Melton West Virginia University Introduction Chebyshev (Tchebycheff) polynomials form a special class of polynomials especially suited for approximating other functions. They are important in approximation theory because the roots of the Chebyshev polynomials are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon (a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equally spaced interpolation points) and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm ( [1], p. 3). Chebyshev Polynomials The Chebyshev polynomial of degree n is defined by: Tn (x) = cos[n arcos(x)], x ε [-1, 1], n = 0, 1, 2, … The Error in Polynomial Interpolation Theorem: If a function f is continuous on [a, b] and (n+1) times differentiable in (a, b), then for any x ε [a, b] there exists a value ξ(x) ε (a, b) , such that: Monic Polynomial Referring back to the properties of the Chebyshev polynomials, we take note that a polynomial of degree n is of the form: θ ε [0, π], n = 0, 1, 2, … The Chebyshev polynomial satisfies the recursive relation: T 0 (x) = 1 T 1 (x) = x Tn+1 (x) = 2 x. Tn (x) – Tn-1 (x), Proof ([3]): Hence, Tn(x) divided by Notice that if x=xk for any k=0: n the result is trivial. Therefore, assume x≠ xk. For a given t we construct the function: Theorem: If pn(x) is a monic polynomial of degree n, then: is monic. Note that is a monic polynomial of degree (n+1), and hence n = 1, 2, … Proof ([1], pg. 4) : Where the constant g(t) is defined such that F(x)=0. F(x) has (n+2) distinct roots in the interval [a, b] which are the nodes x 0, …, xn, x. From the assumption on f it follows that F is a real (n+1) times differentiable function on [a, b]. Now, apply Role’s theorem (n+1) times, it follows that there exists ξ ε [a, b], such that Since Pn is a polynomial of at most degree n, we have for the (n+1)st derivative: Proof ([4]): The minimal value of Suppose that, Supplement And since we can derive: can actually be obtained if . , which is equivalent Let and let xk be the following (n+1) point to choosing xk as the roots of the Chebyshev. polynomial T(n+1)(x). , we have . Hence, Theorem: Tn(x) has n distinct zeros xk that lie in the interval [-1, 1] such that: . , k=0, 1, …, n-1 This means that . Proof ([5]): : function v = tchebycheff ( n, x ) if ( n < 0 ) v = []; return end v = zeros ( 1, n + 1 ); v(1, 1) = 1. 0; if ( n < 1 ) return end x = x(: ); v(1, 2) = x(1, 1); for i = 2 : n v(1, i+1) = 2. 0 * x(1, 1). * v(1, i) - v(1, i-1); end v=v(n+1); return end , we set Since ([2]) If the interpolation points x 0, …. , xn ε [-1, 1]then there exists ξ(n) ε (-1, 1) such that the distance between the function whose values we interpolate, f(x), and the interpolation polynomial, Qn(x) is: + lower degree terms Furthermore, using the equivalent property, Tn (cosθ) = cos(n θ), Chebyshev Nodes Supplement Hence, the polynomial (qn-pn)(x) oscillates (n+1) times in the interval [-1, 1], which means that (qn-pn)(x) has at least n distinct roots in the interval. However, pn(x) and qn(x) are both monic polynomials which means that their difference is a polynomial of degree n-1 at most. Such a polynomial cannot have more than n-1 distinct roots, which leads to a contradiction. Hence, when polynomial interpolation is carried out on the zeros of T(n+1)(x) , the resulting error term is: Acknowledgements Linear spaced points Chebyshev nodes [1]. Satoglu, Humeyra and Gul Balcik. Chebyshev Polynomials, 21 May 2012. Web. 22 April 2014 [2]. Burkardt, John. t_polynomial. fsu. edu, 26 March 2012. Web. 22 April 2014 [3]. Behrens, J. Numerical Analysis I – Proof of Interpolation Error Theorem. Center for Mathematical Sciences, Winter 2005/2006. Web. 22 April 2014 [4]. Levy, D. Interpolation. umd. edu. Web. 22 April 2014 [5]. Chebyshev Expansions. ” Society for Industrial and Applied Mathematics, 2007. Web. 22 April 2014 Moreover, this is the best upper bound that can be achieved by varying the choice of xk.
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