Computation of Chebyshev Polynomials for Union of Intervals
It provides a practical computational method for a specialized problem in approximation theory, but the approach is incremental and domain-specific.
This paper presents numerical procedures based on semidefinite programming to compute Chebyshev polynomials of the first and second kind for a finite union of compact intervals, enabling efficient approximation on disconnected domains.
Chebyshev polynomials of the first and second kind for a set K are monic polynomials with minimal L $\infty$-and L 1-norm on K, respectively. This articles presents numerical procedures based on semidefinite programming to compute these polynomials in case K is a finite union of compact intervals. For Chebyshev polynomials of the first kind, the procedure makes use of a characterization of polynomial nonnegativity. It can incorporate additional constraints, e.g. that all the roots of the polynomial lie in K. For Chebyshev polynomials of the second kind, the procedure exploits the method of moments. Key words and phrases: Chebyshev polynomials of the first kind, Chebyshev polynomials of the second kind, nonnegative polynomials, method of moments, semidefinite programming.