On generalized convolution operators

1972 ◽  
Vol 23 (1) ◽  
pp. 58-69
Author(s):  
Z. Zieleźny
Keyword(s):  
Convolution Operators ◽  
Generalized Convolution

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roland Duduchava
Keyword(s):  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Beltrami Equation ◽  
Bessel Potential ◽  
Convolution Operators ◽  
Classical Setting ◽  
Type Boundary ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 in the non-classical setting when solutions are sought in the Bessel potential spaces \mathbb{H}^{s}_{p}(\mathcal{C}), \frac{1}{p}<s<1+\frac{1}{p}, 1<p<\infty. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain \Omega_{\alpha}\subset\mathbb{R}^{2} of magnitude 𝛼. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J.27 (2020), 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2\times 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes \mathbb{R}^{+} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys.60 (2013), 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl.443 (2016), 2, 707–731].

Transference on some non-convolution operators from euclidean spaces to torus

2010 ◽  
Vol 32 (1) ◽  
pp. 59-68
Author(s):  
Yandan Zhang ◽  
Dashan Fan ◽  
Jiecheng Chen
Keyword(s):  
Convolution Operators ◽  
Euclidean Spaces

scholarly journals Generalized convolution theorem associated with fractional Fourier transform

2012 ◽  
Vol 14 (13) ◽  
pp. 1340-1351 ◽  
Author(s):  
Jun Shi ◽  
Xuejun Sha ◽  
Xiaocheng Song ◽  
Naitong Zhang
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Convolution Theorem ◽  
Generalized Convolution

Sums of Convolution Operators

1972 ◽  
Vol 3 (2) ◽  
pp. 380-388 ◽  
Author(s):  
Stan K. Kranzler
Keyword(s):  
Convolution Operators

UNIFORMLY ASYMPTOTIC SOLUTIONS FOR PSEUDODIFFERENTIAL EQUATIONS WITH SINGULAR INTEGRAL OPERATORS

2001 ◽  
Vol 09 (02) ◽  
pp. 495-513 ◽  
Author(s):  
A. HANYGA ◽  
M. SEREDYŃSKA
Keyword(s):  
Asymptotic Solution ◽  
Frequency Domain ◽  
Singular Integral ◽  
Hyperbolic Equations ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Asymptotic Solutions ◽  
Convolution Operators ◽  
Pseudodifferential Equations

Uniformly asymptotic frequency-domain solutions for a class of hyperbolic equations with singular convolution operators are derived. Asymptotic solutions for this class of equations involve additional parameters — called attenuation parameters — which control the smoothing of the wavefield at the wavefront. At caustics the ray amplitudes have a singularity associated with vanishing of ray spreading and with divergence of an integral controlling the rate of exponential amplitude decay. Both problems are resolved by applying a generalized Kravtsov–Ludwig formula derived in this paper. A different asymptotic solution is constructed in the case of separation of dispersion and focusing effects.

Sign in / Sign up

Export Citation Format

Share Document