scholarly journals Some extensions of a certain integral transform to a quotient space of generalized functions

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Shrideh K.Q. Al-Omari ◽  
Jafar F. Al-Omari
Keyword(s):  
Quotient Space ◽  
Integral Transform ◽  
Generalized Functions ◽  
Space Of Generalized Functions

AbstractIn this paper, we establish certain spaces of generalized functions for a class of ɛ

scholarly journals ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

2020 ◽  
Vol 8 (2) ◽  
pp. 24-39
Author(s):  
V. Gorodetskiy ◽  
R. Kolisnyk ◽  
O. Martynyuk
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Initial Condition ◽  
Partial Derivatives ◽  
Time Problem ◽  
The Cauchy Problem ◽  
Space Of Generalized Functions ◽  
Derivatives Of

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

An integral transform of generalized functions. II

1997 ◽  
Vol 23 (1) ◽  
pp. 115-126
Author(s):  
Anil Kumar Mahato ◽  
Анил Кумар махато
Keyword(s):  
Integral Transform ◽  
Generalized Functions

A Mixed Parseval's Equation And a Generalized Hankel Transformation of Distributions

1989 ◽  
Vol 41 (2) ◽  
pp. 274-284 ◽  
Author(s):  
J. J. Betancor
Keyword(s):  
Dual Space ◽  
Integral Transform ◽  
Generalized Functions ◽  
Generalized Function ◽  
Hankel Transformation ◽  
Formula One ◽  
Image Position ◽  
Complex Valued

Let an integral transform T﹛f﹜ of a complex valued function f(x) defined over the interval (0, ∞) be defined as One of the most usual procedures to extend the classical transform (l.a) to generalized functions consists in constructing a space A of testing functions over (0, ∞) which is closed with respect to the classical transform (l.a) and then the corresponding transform of the generalized function/ of the dual space of A is defined through This approach has been followed by L. Schwartz [13] and A. H. Zemanian [20], amongst others.

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