scholarly journals MULTIVARIATE SAMPLING THEOREMS ASSOCIATED WITH MULTIPARAMETER DIFFERENTIAL OPERATORS

2005 ◽  
Vol 48 (2) ◽  
pp. 257-277 ◽  
Author(s):  
M. H. Annaby
Keyword(s):  
Lagrange Interpolation ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Sampling Theory ◽  
Sampling Theorem ◽  
Second Order ◽  
Two Dimensional ◽  
Single Variable ◽  
Sampling Theorems ◽  
Two Parameter

AbstractWe investigate the multivariate sampling theory associated with multiparameter eigenvalue problems. A several-variable counterpart of the classical sampling theorem of Whittaker, Kotel’nikov and Shannon is given. It arose when the multiparameter system has order one. Two-dimensional sampling theorems associated with two-parameter systems of second-order differential operators will be established. The sampling formulae are of multivariate non-uniform Lagrange interpolation type. Unlike many of the known formulae, the interpolating functions are not necessarily products of single variable functions.

scholarly journals On the eigenvalue problems for differential operators with coupled boundary conditions

2010 ◽  
Vol 15 (4) ◽  
pp. 493-500 ◽  
Author(s):  
S. Sajavičius
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Second Order ◽  
Two Dimensional ◽  
Complex Eigenvalues ◽  
Coupled Boundary Conditions ◽  
Analytical Expressions

In the paper, the eigenvalue problems for one- and two-dimensional second order differential operators with nonlocal coupled boundary conditions are considered. Conditions for the existence of zero, positive, negative or complex eigenvalues are proposed and analytical expressions of eigenvalues are provided.

scholarly journals Multidimensional sampling theorems for multivariate discrete transforms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. A. Hassan
Keyword(s):  
Spectral Properties ◽  
Lagrange Interpolation ◽  
Orthonormal Basis ◽  
Difference Operator ◽  
Two Dimensional ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Sampling Theorems ◽  
Discrete Transforms ◽  
Discrete Type

AbstractThis paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green’s function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings.

scholarly journals On sampling expansions of Kramer type

2004 ◽  
Vol 2004 (5) ◽  
pp. 371-385
Author(s):  
Anthippi Poulkou
Keyword(s):  
Boundary Value Problems ◽  
Discrete Spectrum ◽  
Differential Operators ◽  
Boundary Value ◽  
Sampling Theorem ◽  
Second Order ◽  
Joint Work ◽  
First Order ◽  
Survey Paper ◽  
Sampling Expansions

We treat some recent results concerning sampling expansions of Kramer type. The linkof the sampling theorem of Whittaker-Shannon-Kotelnikov with the Kramer sampling theorem is considered and the connection of these theorems with boundary value problems is specified. Essentially, this paper surveys certain results in the field of sampling theories and linear, ordinary, first-, and second-order boundary value problems that generate Kramer analytic kernels. The investigation of the first-order problems is tackled in a joint work with Everitt. For the second-order problems, we refer to the work of Everitt and Nasri-Roudsari in their survey paper in 1999. All these problems are represented by unbounded selfadjoint differential operators on Hilbert function spaces, with a discrete spectrum which allows the introduction of the associated Kramer analytic kernel. However, for the first-order problems, the analysis of this paper is restricted to the specification of conditions under which the associated operators have a discrete spectrum.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Sign in / Sign up

Export Citation Format

Share Document