The convergence rates of expansions in Jacobi polynomials

Summary

We consider the convergence of expansions of the form

$$f\left( \chi \right) = \sum\limits_{n = 0}^\infty {b_n A_n P_n^{\left( {\alpha ,\beta } \right)} } \left( \chi \right)$$

whereA _n P _n ^(α,β) is then ^th normalised Jacobi polynomial. A modification of a technique described previously by Mead and Delves (1973) provides estimates of the convergence rate of the coefficientsb _n which are more powerful than those so far available. The methods used are applicable to other orthogonal expansions, and the results obtained here are used in the companion paper (Delves and Bain (1976)) to discuss the optimum choice of weight functions in a class of variational methods.