Representation of the solution of a homogeneous convolution equation in the form of a series

1991 ◽  
Vol 49 (3) ◽  
pp. 259-262
Author(s):  
T. T. Kuzbekov
Keyword(s):  
Convolution Equation ◽  
Homogeneous Convolution Equation

The Schauder basis in the solution space of a homogeneous convolution equation

1995 ◽  
Vol 57 (1) ◽  
pp. 41-50 ◽  
Author(s):  
A. S. Krivosheev
Keyword(s):  
Solution Space ◽  
Convolution Equation ◽  
Schauder Basis ◽  
Homogeneous Convolution Equation

Some results in spectral synthesis

1965 ◽  
Vol 61 (2) ◽  
pp. 395-424 ◽  
Author(s):  
Robert J. Elliott
Keyword(s):  
Spectral Synthesis ◽  
Related Result ◽  
Convolution Equation ◽  
Continuous Functions ◽  
Exponential Polynomial ◽  
Space Of Continuous Functions ◽  
Translation Invariant ◽  
Famous Paper ◽  
Translation Invariant Subspace ◽  
Homogeneous Convolution Equation

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

scholarly journals Solvability Criterion for Convolution Equations on a Half-Strip

2021 ◽  
Vol 15 (4) ◽  
Author(s):  
Volodymyr Dilnyi
Keyword(s):  
Information Theory ◽  
Spectral Analysis ◽  
Hardy Space ◽  
Convolution Equation ◽  
Weighted Hardy Space ◽  
Decomposition Property ◽  
Half Strip ◽  
Solvability Criterion ◽  
Convolution Equations ◽  
Homogeneous Convolution Equation

AbstractWe obtain the criterion of solvability of homogeneous convolution equation in a half-strip. Proof is based on a new decomposition property of the weighted Hardy space. This result has relations to the spectral analysis-synthesis problem, cyclicity problem, information theory. All data generated or analysed during this study are included in this published article.

Fluctuations of random walk in R d and storage systems

1977 ◽  
Vol 9 (03) ◽  
pp. 566-587 ◽  
Author(s):  
Priscilla Greenwood ◽  
Moshe Shaked
Keyword(s):  
Random Walk ◽  
Unique Solution ◽  
Convex Cone ◽  
Storage Systems ◽  
Queueing Systems ◽  
Convolution Equation ◽  
Stopping Times ◽  
Multivariate Distributions ◽  
Explicit Formulas ◽  
And Storage

Two Wiener-Hopf type factorization identities for multivariate distributions are introduced. Properties of associated stopping times are derived. The structure that produces one factorization also provides the unique solution of the Wiener-Hopf convolution equation on a convex cone in R d . Some applications for multivariate storage and queueing systems are indicated. For a few cases explicit formulas are obtained for the transforms of the associated stopping times. A result of Kemperman is extended.

Approximation theorem for a homogeneous vector convolution equation

2004 ◽  
Vol 195 (9) ◽  
pp. 1271-1289
Author(s):  
I F Krasichkov-Ternovskii
Keyword(s):  
Approximation Theorem ◽  
Convolution Equation ◽  
Homogeneous Vector
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