Fourier expansion associated with the Lorentz group in the space of generalized functions with support in the light cone

1978 ◽  
Vol 34 (1) ◽  
pp. 14-21
Author(s):  
Yu. G. Shondin
Keyword(s):  
Lorentz Group ◽  
Light Cone ◽  
Fourier Expansion ◽  
Generalized Functions ◽  
Space Of Generalized Functions

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.

scholarly journals On Analog of Fourier Transform in Interior of the Light Cone

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Tatyana Shtepina
Keyword(s):  
Fourier Transform ◽  
Lorentz Group ◽  
Light Cone ◽  
Inverse Transform ◽  
Irreducible Components ◽  
One To One

We introduce an analog of Fourier transformFhρin interior of light cone that commutes with the action of the Lorentz group. We describe some properties ofFhρ, namely, its action on pseudoradial functions and functions being products of pseudoradial function and space hyperbolic harmonics. We prove thatFhρ-transform gives a one-to-one correspondence on each of the irreducible components of quasiregular representation. We calculate the inverse transform too.

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