From high oscillation to rapid approximation II: Expansions in Birkhoff series

We consider the use of eigenfunctions of polyharmonic operators, equipped with homogeneous Neumann boundary conditions, to approximate nonperiodic functions in compact intervals. Such expansions feature a number of advantages in comparison with classical Fourier series, including uniform convergence and more rapid decay of expansion coefficients. Having derived an asymptotic formula for expansion coefficients, we describe a systematic means to find eigenfunctions and eigenvalues. Next we demonstrate uniform convergence of the expansion and give estimates for the rate of convergence. This is followed by the introduction and analysis of Filon-type quadrature techniques for rapid approximation of expansion coefficients. Finally, we consider special quadrature methods for eigenfunctions corresponding to a multiple zero eigenvalue.