Extreme problem for operator with two kernels in spaces of generalized functions

Keywords: extremal problem, solution in quadratures, operators of convolution type, operators with two kernels, Fourier transform, Riemann problem of the theory of analytic functions, generalized functions


The article considers an extremal problem for an operator with two kernels in spaces of generalized functions. It is emphasized that in the absence of "regular" asymptotics of the modules of zeros (even if they all lie on the same ray), the behavior of the function is difficult to predict. In a situation where some restrictions are set on the zeros of the studied subclass of entire functions, but there is no "correct" distribution of zeros, only extreme problems of certain asymptotic characteristics of the functions of the given class can be solved. It is emphasized that the relevance of the consideration of generalized functions is due to their numerous applications in such important sections of complex analysis as the problems of completeness of exponential systems, the theory of interpolation, the theory of analytical continuation. The extreme problem of finding a function for an operator with two kernels in the spaces of generalized functions of the Sobolev-Slobodetsky type under appropriate conditions is presented. It is emphasized that everything else, except for the given one, is given in the problem. The solution of the problem is presented in squares. The solution is written in terms of projection operators and Fourier transform operators.

Denoting the expression under the modulus and taking into account the solvability conditions of the equation in convolutions with two kernels, two problems are formed: the extremal problem and the problem of determining the function under the appropriate conditions. Where is denoted in a certain way the set of functions for which the equation with two kernels is solvable subject to the additional conditions of problem 2.

The paper proves that problem 1 has a unique solution, this solution was found analytically. Problem 2 is solvable. In Fourier representations, it is reduced to the Riemann problem in generalized functions. The solution depends significantly on the value of the index of the equation with two kernels. If the index is less than or equal to zero, then the extremal problem has a unique solution. If the index is greater than zero, then the solution of the Riemann problem is not unique and depends on arbitrary complex constants. This solution must also satisfy the additional conditions of problem 2. A system of equations with unknowns is obtained. It is proved that if the rank of this system is less than the index, then the general solution of the extremal problem depends on the result of subtracting the rank of arbitrary complex constants from the index


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How to Cite
Hryhoriev, Y. (2023). Extreme problem for operator with two kernels in spaces of generalized functions. COMPUTER-INTEGRATED TECHNOLOGIES: EDUCATION, SCIENCE, PRODUCTION, (52), 60-70. https://doi.org/10.36910/6775-2524-0560-2023-52-08
Computer science and computer engineering