An improved upper bound for ultraspherical coefficients

Document Type : Research Article


1 Department of Mathematics, Faculty of Science and Modern Technology, Graduate University of Advanced Technology, Kerman, Iran

2 Department of Applied Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran


In this paper, new upper bounds for the ultraspherical coefficients of differentiable functions are presented. Using partial sums of ultraspherical polynomials, error approximations are presented to estimate differentiable functions. Also, an error estimate of the Gauss-Jacobi quadrature is obtained and we state an upper bound for Legendre coefficients which is sharper than upper bounds proposed so far. Numerical examples are given to assess the efficiency of the presented theoretical results.