We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with high-order filter polynomials obtained from a regularized Chebyshev expansion of a window function. After a short discussion of the conceptual foun...

The usual way to determine the asymptotic behavior of the Chebyshev coefficients for a function is to apply the method of steepest descent to the integral representation of the coefficients. However, the procedure is usually laborious. We prove an asymptotic upper bound on the Chebyshev coefficients for the kth integral of a function. The tightness of this upper bound is then analyzed for the c...

In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: polynomial and generalized rational approximation. the second case, approximations are ratios of linear forms basis functions not limited monomials. It is already known that in case grid corresponding optimization problems can be reduced solving programming ...

1. Introduction and Summary. Clenshaw [1, 2] has described a simple and effective method for summing a Chebyshev series based on the recurrence relation between the Chebyshev coefficients. He has further pointed out that the same method could be used to sum a series of any set of functions which are generated by a linear recurrence relation [1]. Such a set of functions is a set of orthogonal po...

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...

In a recent article, the first and second kinds of multivariate Chebyshev polynomials fractional degree, relevant integral repesentations, have been studied. this we introduce pseudo-Lucas functions show possible applications these new functions. For kind, compute Newton sum rules any orthogonal polynomial set starting from entries Jacobi matrix. representation formulas for powers r×r matrix, a...

A unified approach is presented for determining all the constants Ym.n (m > 0, n > 0) which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that Ym,m+2 = 1/3 (m > 0), a problem which had remained open.