Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities

Abstract

In this paper, new and optimal asymptotics on the decay of the coefficients for functions of limited regularity expanded in terms of Jacobi and Gegenbauer polynomial series are presented. For a class of functions with interior singularities, the decay of the coefficient is of the same asymptotic order for arbitrary $$\alpha ,\,\beta >-1$$, which confirms that the decay of the coefficients in the Jacobi polynomial series without normalization is a factor of $$ \sqrt{n}$$ slower compared with the Chebyshev expansion. While for functions with boundary singularities, the decay depends on $$\alpha $$ and $$\beta $$ with $$\alpha ,\,\beta >-1$$. For Gegenbauer expansion, it is related to the parameter $$\lambda $$ whatever f with interior or boundary singularities. All of these asymptotic analysis are optimal. Moreover, under the optimal asymptotic analysis, it derives that the truncated spectral expansions with some specific parameters can achieve the optimal convergence rates, i.e., the same as the best polynomial approximation in the sense of absolute maximum error norm. Numerical examples illustrate the perfect coincidence with the estimates.