Solution to the Modified Helmholtz Equation for Arbitrary Periodic Charge Densities

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.

Corrected Fourier series and its application to function approximation

Any quasismooth functionf(x)in a finite interval[0,x0], which has only a finite number of finite discontinuities and has only a finite number of extremes, can be approximated by a uniformly convergent Fourier series and a correction function. The correction function consists of algebraic polynomials and Heaviside step functions and is required by the aperiodicity at the endpoints (i.e.,f(0)≠f(x0)) and the finite discontinuities in between. The uniformly convergent Fourier series and the correction function are collectively referred to as the corrected Fourier series. We prove that in order for themth derivative of the Fourier series to be uniformly convergent, the order of the polynomial need not exceed(m+1). In other words, including the no-more-than-(m+1)polynomial has eliminated the Gibbs phenomenon of the Fourier series until itsmth derivative. The corrected Fourier series is then applied to function approximation; the procedures to determine the coefficients of the corrected Fourier series are illustrated in detail using examples.

On Grasia's criterion for uniform convergence of Fourier series

Garsia's discovery that functions in the periodic Besov space λ(p-1,p, 1), with 1 <p< ∞, have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in λ(p-1,p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to λ(α,p, q)