The Law of Transforming the Coefficients of a Trigonometric Fourier Series under the Mapping of a Continuous Function into a Discontinuous Function of the First Kind

Abstract

In this paper, we describe some properties of an infinite-dimensional linear Hilbert space of 2π-periodic functions with a scalar product defined in it. This space is constructed over a set of real numbers. In particular, the law of transformation of the coefficients of the trigonometric Fourier series is defined in the case of transformation of a continuous function on the period into a discontinuous function of the first kind. In this regard, strictly justified and obtained expressions that in analytical form disclose the said mathematical law. A successful solution of the problem became possible as a result of the author introducing an auxiliary system basis of orthogonal functions. This made it possible to obtain an intermediate expansion of each of the original Fourier coefficients of a continuous periodic function, and with the subsequent rearrangement of the terms it was possible to form the decomposition of a discontinuous periodic function and determine the Fourier coefficients of such decomposition. This approach has revealed the compositional nature of the required mathematical law of transformation. This mathematical problem appears and becomes relevant during the spectral analysis of continual physical and technical dynamic systems in the case of discretization of their motion in space and time. As an example of such systems, electrical or radio engineering systems can be called. The solution of the problem makes it possible to supplement the spectral (frequency) method of analyzing electrical circuits at the present time with missing mathematical models that are able to directly and without resorting to the integration operation to identify and evaluate the distribution of the spectral components of the trigonometric Fourier series in the case of discretization of the continuous motion of dynamical systems. The above, for example, can be attributed to devices and systems power semiconductor electronics in which the generation, transportation and conversion of electrical (electromagnetic) energy is carried out by the method of discretizing the energy processes that are continuous in their physical nature. Today, this approach has become a paradigm in the development of modern power electronics.