Effective solution of a linear system with Chebyshev coefficients

This paper presents an efficient algorithm for a special triangular linear system with Chebyshev coefficients. We present two methods of derivations, the first is based on formulae where the nth power of x is solved as the sum of Chebyshev polynomials and modified for a linear system. The second deduction is more complex and is based on the Gauss–Banachiewicz decomposition for orthogonal polynomials and the theory of hypergeometric functions which are well known in the context of orthogonal polynomials. The proposed procedure involves 𝒪(n m) operations only, where n is matrix size of the triangular linear system L and m is number of the nonzero elements of vector b. Memory requirements are 𝒪(m), and no recursion formula is needed. The linear system is closely related to the optimal pulse-wide modulation problem.