Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of [formula omitted

Abstract

If the coefficients in a Fourier cosine series, f ( x ) ≈ f N = ∑ n = 0 ∞ a n cos ( nx ) , decrease as a small negative power of n, then one may need millions of terms to sum the series to high accuracy. We show that if the a n are known analytically and have a power series in 1 / n , then it is straightforward to approximate f ( x ) as a series of what we shall the Lanczos–Krylov (LK) functions. (We describe the similar methodology for sine series; general Fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled, too.) For cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n, the LK functions may be expressed in terms of Bernoulli polynomials. The LK functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly; these are also known as “Clausen functions”. We provide rapidly convergent series to compute these Clausen functions to high accuracy. Our method includes the “endpoint subtraction” ideas of Lanczos and Krylov, but is more general. The sum ∑ n = 1 ∞ ( ± 1 ) n + 1 ( 1 / ( n + λ ) ) cos ( nx ) , where λ > 0 is a constant, arises in phase transitions in absorbed monolayers on metal surfaces. It is easily summed by our method, which correctly incorporates the logarithmic singularities at x = ± π .