Quadratic exponents with applications: finite sums, series, integral transforms

We shall talk over various applications in mathematics of quadratic exponents (exponential functions with arguments quadratic in variables). We shall talk over various applications in mathematics of quadratic exponents (exponential functions with arguments quadratic in variables). As an introduction, the most striking applications, both long-standing and relatively modern, are listed; for example, quadratic Gauss sums, Fraunhofer experiment and Fresnel integrals, the Gaussian package which is the first and only computer program whose creators were awarded the Nobel Prize for solving a number of problems of previously inaccessible complexity using this package. Further, our talk consists of three parts, in accordance with the title. The first part will talk about quadratic Gauss sums, the history of their calculation and some generalizations and applications. The second part lists some series related to quadratic exponentials, including the Jacobi theta functions. The problems of expansion of functions in terms of a system of integer shifts of Gaussian functions (quadratic exponentials!) presented, and brief information about the theory of coherent states and its applications in quantum physics and optics is given. The results in this direction, obtained including by the author of the report and with his participation, presented in more detail. The third part sets out the basic facts of the theory of quadratic Fourier transforms, a many-sided transformation that has many alternative names, including the “fractional Fourier transform” and a number of others. The lecture has a mathematical focus; at the same time, facts from the history of mathematics related to the concepts introduced and the mathematicians who studied them are given.
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