scholarly journals Multidimensional Tauberian theorems for vector-valued distributions

2014 ◽  
Vol 95 (109) ◽  
pp. 1-28 ◽  
Author(s):  
Stevan Pilipovic ◽  
Jasson Vindas
Keyword(s):  
Asymptotic Stability ◽  
Integral Transform ◽  
Tauberian Theorems ◽  
Sufficient Condition ◽  
Cauchy Problems ◽  
Necessary And Sufficient Condition ◽  
The Laplace Transform ◽  
Distribution Spaces ◽  
Necessary And Sufficient ◽  
Vector Valued

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform Mf?(x,y) = (f*?y)(x), (x,y) ? Rn ? R+, with kernel ?y(t) = y?n?(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on {x0}?Rm. In addition, we present a new proof of Littlewood?s Tauberian theorem.

scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

1976 ◽  
Vol 74 ◽  
pp. 71-79
Author(s):  
R. Datko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

Vector-valued nonuniform multiresolution analysis on local fields

2015 ◽  
Vol 13 (04) ◽  
pp. 1550029 ◽  
Author(s):  
Firdous Ahmad Shah ◽  
M. Younus Bhat
Keyword(s):  
Multiresolution Analysis ◽  
Orthonormal Basis ◽  
Positive Characteristic ◽  
Local Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vilenkin Groups ◽  
Refinement Mask ◽  
Necessary And Sufficient ◽  
Vector Valued

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

scholarly journals Lyapunov-Krasovskii stability analysis of nonlinear integro-differential equation

2018 ◽  
Vol 7 (2) ◽  
pp. 53
Author(s):  
Prebo Jackreece
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Integro Differential Equation ◽  
Necessary And Sufficient

The purpose of this paper is to develop a qualitative stability analysis of a class of nonlinear integro-differential equation within the framework of Lyapunov-Krasovskii. We show that the existence of a Lyapunov-Krasovskii functional is a necessary and sufficient condition for the uniform asymptotic stability of the nonlinear Volterra integro-differential equations.

A New Procedure for Designing a Kind of Orthogonal Vector-Valued Multiscaled Wavelets and Wavelet Wraps

2011 ◽  
Vol 204-210 ◽  
pp. 1733-1736
Author(s):  
Hong Wei Gao
Keyword(s):  
Multiresolution Analysis ◽  
Filter Bank ◽  
Matrix Theory ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthogonal Vector ◽  
Novel Method ◽  
Necessary And Sufficient ◽  
Vector Valued

In this paper, notion of vector-valued multiresolution analysis is introduced. So does the notion of orthogonal vector-valued wavelets A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is presented by using paraunitary vector filter bank theory and matrix theory. A novel method for constructing a kind of orthogonal shortly supported vector -valued wavelets is presented.

An Optimization Algorithm for Orthogonal Trivariate Wavelets with Short Support

2012 ◽  
Vol 461 ◽  
pp. 835-839
Author(s):  
Ke Zhong Han
Keyword(s):  
Filter Bank ◽  
Scaling Function ◽  
Wavelet Packets ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthogonal Vector ◽  
Multi Scale ◽  
Dilation Factor ◽  
Necessary And Sufficient ◽  
Vector Valued

Wavelet analysis is nowadays a widely used tool in applied mathe-matics. The advantages of wavelet packets and their promising features in various application have attracted a lot of interest and effort in recent years.. The notion of vector-valued binary wavelets with two-scale dilation factor associated with an orthogonal vector-valued scaling function is introduced. The existence of orthogonal vector-valued wavelets with multi-scale is discussed. A necessary and sufficient condition is presented by means of vector-valued multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal vector-valued wave-lets with compact support is proposed, and their properties are investigated.

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