New inversion formulas for the integral transformations of Laplace, Weierstrass and Mellin

Background. Analytical methods for solving problems in mathematical physics, including integral transformations, are an actively developing field of mathematical modeling. The method of integral transformations is one of the most effective analytical methods for solving model problems of mathematical physics. In addition to direct applications in physics and in solving boundary value problems of mathematical physics, integral transformations arise in technology for encoding and filtering signals. The currently available inversion formulas for the Laplace, Weierstrass, and Mellin integral transformations have a serious drawback: they require the complex domain or contain derivatives of arbitrarily large order. Both of these drawbacks lead to computational problems. To solve them, we prove new formulas for the direct and inverse integral Fourier transforms, the two-sided integral Laplace transform, the Weierstrass integral transforms, and Mellin transforms. The new formulas do not contain derivatives and are obtained in the form of a series оn the system of orthogonal Hermite polynomials. Their applications in the theory of signal filtering are found. Materials and methods. The work is based on the theoretical positions of the Fourier analysis and Hermite series theory; the expansion theorems for the integral Laplace, Weierstrass, and Mellin transformations are used. Results. We obtain new inversion formulas for the Weierstrass integral transformation by expanding the kernels of the integral representation in a series in the Hermite polynomials. Further, we establish inversion formulas for other integral transformations by using the formulas for the connection of the integral Laplace, Mellin, and Weierstrass transformations. Conclusions. The new inversion formulas for integral transforms that have been established in the paper open up the previously unknown possibilities of applying the classical methods of Fourier, Laplace, Weierstrass, Mellin integral transforms in the theory of signal filtering and in the theory of inverse mathematical physics problems.