Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Abstract

Rayleigh functions σ l (ν) are defined as series in inverse powers of the Bessel function zeros λ ν,n = 0, σ l (ν) = ∞ n=1 1 λ 2l ν, n , where l = 1, 2, . . . ; ν is the index of the Bessel function J ν (x) and n = 1, 2, . . . is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, R l (m) = ∞ k=−∞ σ l |m − k| σ l |k| for l = 1, 2, . . . ; m = 0, ±1, ±2, . . . , are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699-725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424], where the properties of R 1 (m) were investigated.In the present work a general representation of R l (m) in terms of σ l (ν) is deduced. On the basis of this a representation for the function R 2 (m) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R 2 (m) as |m| → ∞. Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded ✩ domains with periodicity conditions in one coordinate. As an example of application of R l (m) a forced Boussinesq equation u tt − 2bΔu t = −αΔ 2 u + Δu + βΔ u 2 + f with α, b = const > 0 and β = const ∈ R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R 1 (m) and R 2 (m) which are responsible for the nonlinear smoothing effect.

Rayleigh based bilateral filtering for flash and no-flash image pairs

“…Thus, the underlying principle is to use a different decaying function to achieve smoothing in the domain filter; rather than the Gaussian function [30]- [31]. Rayleigh distribution function can be regarded as both real and imaginary parts of the complex signal which is corrupted with zero mean uncorrelated Gaussian noise.…”

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

“…(25) in Wu et al [29, p. 6]. Varlamov [26,27] systematically investigated convolutions of the Rayleigh functions with respect to the Bessel index and attempted to exhibit their usefulness for constructing global-in-time solutions of semi-linear evolution equations in circular domains.…”

“…v being the index of the Bessel function J v (x) whose zeros are λ v,n . Varlamov [26,27] succeeded in treating the Rayleigh functions σ l (v) as special functions by presenting some references for functions of the Rayleigh type as well as in their own right (see the references cited by Varlamov [26,27]).…”

“…Varlamov [27] also presented a general formula for R l (m) in terms of σ l (v) and expressed R 2 (m) in terms of the ψ-function, together with its asymptotic formula as |m| → ∞, by making use of the ψ-function, as follows: expression:…”

“…Furthermore, Varlamov [27] provided an example of an initial-boundary-value problem leading to an explicit appearance of R 1 (m) and R 2 (m). In order to show a usefulness of the polygamma functions, we express R 3 (m) in terms of the polygamma functions and give its asymptotic formula as |m| → ∞.…”