Computational Techniques Based on the Lanczos Representation

In his book Discourse on Fourier Series, Lanczos deals in some detail with representations of $f(x)$ of the type $f(x) = {h_{p - 1}}(x) + {g_p}(x)$ where ${h_{p - 1}}(x)$ is a polynomial of degree $p - 1$ and ${g_p}(x)$ has the property that its full range Fourier coefficients converge at the rate ${r^{ - p}}$. In Part I, some properties of ${h_p}(x)$ and of the series $\{ {h_p}(x)\} _1^\infty$ are described. These properties are used here to provide criteria for the convergence or divergence of the Euler-Maclaurin series, in the case when $f(x)$ is an analytic function. The similarities and differences between this series and the Lidstone and other two-point series are briefly mentioned. In Part II, the Lanczos representation is employed to derive an approximate representation $F(x)$ for an analytic function $f(x)$ on the interval [0, 1] is derived. This has the form \[ F(x) = \sum \limits _{q = 1}^{p - 1} {{\lambda _{q - 1}}{B_q}(x)/q! + 2\sum \limits _{r = 0}^{m/2} {({\mu _r}\cos 2\pi rx + {\nu _r}\sin 2\pi rx)} } \] and requires for its determination the values of the derivatives ${f^{(q - 1)}}(1) - {f^{(q - 1)}}(0)\;(q = 1,2, \cdots p - 1)$ and the regularly spaced function values $f(j/m)\;(j = 0,1, \cdots ,m)$. It involves replacing $g_p(x)$ by a discrete Fourier expansion based on trapezoidal rule approximations to its Fourier coefficients. This representation is a powerful one. The drawback is that it requires derivatives. Most of Part II is devoted to the effect of using only approximate derivatives. It is shown that when these are successively less accurate with increasing order (the sort of behaviour encountered using finite difference formula), then the representation is still powerful and reliable. In a computational context the only penalty for using inaccurate derivatives is that a larger value of m may—or may not—be required to attain a specific accuracy.