This paper introduces a new set of exchange rules for a Remez-like algorithm for the Chebyshev design of IIR digital lters. It is explained that the essential di culty, in applying the Remez algorithm to rational functions, is that on some iteration, there may be no solution to the interpolation problem for which the denominator is strictly non-zero in the interval of approximation. Then the us...

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, Tk(x) and Uk(x). In our normalization, the indices are allowed to be halfintegers as well as integers, and the “polynomials” actually live in Z[x, √ 2− x, √ 2 + x]. We will show that the rational functions Tm/2(x) Tn/2(x) and Um/2(x) Un/2(x) are very remarkable from the point of view of G...

We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions. Keywords—Approximation Theory, Chebyshev Polynomial, Computable Functions, Comp...

In this manuscript, a numerical technique is presented for finding the eigenvalues of the regular Sturm-Liouville problems. The Chebyshev cardinal functions are used to approximate the eigenvalues of a regular Sturm-Liouville problem with Dirichlet boundary conditions. These functions defined by the Chebyshev function of the first kind. By using the operational matrix of derivative the problem ...