Determination of trigonometric Fourier’s series by the method of four-dimensional mathematics

This paper is devoted to the determination of the trigonometric Fourier series by the method of four-dimensional mathematics. In this paper, a new four-dimensional method for evaluation of the sums containing trigonometric function is proposed. Current work provides a first study and finding towards Fourier’s series in four-dimensional space.


Introduction
In this paper, we study a trigonometric form of Fourier series in four-dimensional mathematics. Skills on trigonometry functions play an important role in a wide variety of careers, including architecture and engineering. By referring to this, it's important for students that are interested in the scientific or engineering fields to understand trigonometry. The purpose of this work is to formulate a new formulation of Fourier's series by using new approach theory of four-dimensional variables. To achieve this goal, it is necessary to study a theory of four-dimensional variables, determination of main formulas in this theory in the shade of the functional analysis, spectral theory and modern methods of mathematical analysis.
The theory of four-dimensional functions is the new method in mathematics and due to this is scantily explored. In 2015 it was possible to describe the initial chapters of this theory by Kazakh researcher M.M.Abenov [1][2]. The discovery of the spectral theory made it possible to systematize the available results and obtain new information about theory of four-dimensional variables and solved three dimensional problems analytically [2][3]. Full substantiated analysis of this theory was published by Abenov M. (Al-Farabi Kazakh National University). While developing this theory, Abenov M.M. and Gabbasov M.B. found all four-dimensional Abenov spaces with commutative multiplication, which were assigned the designations M2-M7, and it became necessary to study these spaces. In this paper we performed a research in Abenov space of four-dimensional numbers M3 [4][5].
The theory of four-dimensional functions is based on the principles of commonality of its key concepts with similar categories of one-dimensional and two-dimensional analysis [2]. This approach is justified by the fact that linear spaces of one-dimensional and two-dimensional numbers can initially be considered as eigenspaces of a more general space of four-dimensional numbers. This leads to the study IOP Publishing doi:10.1088/1742-6596/1988/1/012027 2 of these one-dimensional, two-dimensional and four-dimensional numbers as elements of a triune system, which allows to uniformly define key concepts in the theory of these numbers (for example, arithmetic operations on the set of these numbers) [2][3][4][5]. Note that a similar approach was previously used by the physicist W. Hamilton when he constructed the quaternion algebra. He defined a non-commutative operation of multiplication of elements, which allowed avoiding zero divisors [2].

Methodology
Let consider a linear space ℜ 4 , which contains vectors with four components. Following the standard convention in linear algebra we shall denote an element ∈ ℜ 4 as a column vector We define addition of the vectors and multiplication of the vector by scalar by components, thus, for = ( Simplex module of the element ∈ ℛ 4 , is said to be a number defined by We define a space Division of ∈ ℛ 4 is defined by multiplying to inverse element. Note that division fulfill only for nongenerate numbers [2]. ) ∈ (4, ℝ) The linear combination will be mapped to linear combination of the corresponding matrices: The multiplication of the basic numbers The vector with the components (2) will be denoted by Λ :  Let denote by Λ = {Λ : ∈ ℛ 4 }. Let define a map : ℛ 4 → Λ, by assigning to each element ∈ ℛ 4 its spectral number Λ ∈ Λ. It is not difficult to show that the map is bijection.
There exists a single regular function satisfying the differential equation: ) ∈ ℛ 4 , then the exponential function can be written as follows [2]: To justify the name of exponential function we need to verify that newly defined function will inherit main properties of the normal exponential function in one dimensional case. First we prove the following formula:

Conclusion
Using the new multiplication rule of the vectors in ℛ 4 we introduce four-dimensional trigonometric functions. By applying the properties of the ℛ 4 we extended the formula for trigonometric sum to the case of ℛ 4 . Obtained formula will be useful in the theory of four-dimensional trigonometric series. Further investigations related to the convergence of the four-dimensional trigonometric series will another paper.