On the Convergence Rate of Generalized Fourier Expansions

We consider the problem of estimating the generalized Fourier coefficients b_nin an expansion of the type $f(x)=∑bnhn(x)$. We give a simple method of obtaining a priori estimates of b_nand present a detailed analysis of the convergence obtained with several frequently employed orthogonal expansion sets. The results depend only on the general analytic structure of the function f(x), and are relevant to a recent discussion of the convergence of variational calculations (Delves & Mead, 1971). A simple extension of the results also gives estimates of the pointwise convergence of the expansions.