On Matrix Representation of Bernstein Polynomials for Triple Sequences

Corresponding Author: Ayhan Esi Department of Basic Engineering Sciences, Malatya Turgut Ozal University, 44040, Malatya, Turkey Email: aesi23@hotmail.com Abstract: In this article, we give an overview of other commonly used basis for the triple sequence space of Bernstein polynomials, the Bernstein basis and put forward a matrix representation of Bernstein polynomials for the triple sequence space and its many useful properties.


Introduction
Proof of the famous Weierstrass Approximation Theorem was found by (Bernstein, 1912). Polynomials, called Bernstein polynomials, are defined with the help of probability theory as follows: Let Later, Bernstein polynomials, bringing them into an intensive field of research were found to have many remarkable features.
In this article, we have studied polynomials that are simply defined, incredibly useful mathematical tools that can be calculated quickly in computer systems and representing a wide variety of functions. Consider that: as a linear combination of certain basic polynomial represents a polynomial {(1, x, x 2 , x 3 ,x n )}. Generally, a polynomial function with a degree less than or equal to n, can be expressed as: i. A set of polynomials equal to or less than n forms a vector space: Polynomials can be added together, multiplied by a scalar and all vector space properties apply ii. (ii) The set of functions {(1, x, x 2 , x 3 ,x n )} forms a basis for this vector space -that is, the polynomial of any degree less than or equal to n is uniquely a linear combination Often called the power basis, this basis is just one of the infinitely many bases for the triple sequence space of Bernstein polynomials. Bernstein polynomials play an important role in other areas of mathematics and approximation theory. They also play an important role in physics. So now we give the definition of these polynomials and their important properties: The Bernstein operator of order rst is given by:  Bernstein polynomials (Brst(f,x)) is said to be statistically convergent to 0  , denoted by st-lim x = 0, provided that the set: has natural density zero for any  > 0. In this case, 0 is called the statistical limit of the triple sequence of Bernstein polynomials. i.e., (K) = 0. That is, Let f be a continuous function defined on the interval [0, 1]. A triple sequence of Bernstein polynomials (Brst(f,x)) is said to be statistically analytic if there is a positive number M such that: A triple sequence (real or complex) can be defined as a function   : x    , where , and represent the set of natural numbers, real numbers and complex numbers respectively. Different types of the triple sequence concept were introduced and first explored by (Sahiner et al., 2007;Sahiner and Tripathy, 2008;Esi, 2014;Esi and Catalbas, 2014;Esi and Savas, 2015;Esi and Sapszoglu, 2012;Esi and Subramanian, 2018b;Esi et al., 2017a;Dutta et al., 2013;Subramanian and Esi, 2015;Esi and Subramanian, 2018a;Velmurugan and Subramanian, 2018;Hazarika et al., 2018;Subramanian et al., 2019;Debnath et al., 2015;Aiyub et al., 2017;Sharma and Esi, 2013), many others.

A Basis of the Bernstein Polynomials of Triple Sequence Spaces
Why is the basically triple sequence of Bernstein polynomials, Bernstein polynomials of order n a basis for the triple sequence of polynomials of order n! equal to or lower? (i) It covers the triple sequence space of polynomialsany polynomial of order less than or equal to n can be written as a linear combination of Bernstein polynomials This easily covers triple sequence space of polynomials and the power basis of any member of the Bernstein polynomials in terms of power can be written as a linear combination of notice (ii) They are linearly independent-that is, there exist constants c0, c1, c2cn so that the identity: holds for all t, then all mnk, i cs  must be zero.
If this is true, we could write:

Bernstein Polynomials for Triple Sequence Spaces
The Bernstein polynomials of degree rst are defined by: for m, n, k = 0,1,2,, r, s, t. There are (r +1) (s +1) (t +1), (rst) th degree Bernstein polynomials for mathematical compatibility we usually set Bmnk,rst = 0 for m, n, k < 0 or m > r, n > s, k > t. It is quite easy to write the coefficients of these polynomials r s t m n k             from Pascal's triangle, the exponents in the x term increase as m, n, k increases and the exponents on the (1-x) term decrease by one as m, n, k increases. In simple cases, we get ( Fig. 1)

Converting the Bernstein Basis to the Power Basis for Triple Sequence Spaces
Since the power basis {1, x, x 2 , x 3 ,x n } forms a basis for the triple sequence of Bernstein polynomials of degree less than or equal to n, any Bernstein polynomial of degree n can be written in term of the power basis. This direct Bernstein polynomial can be calculated using the following definition of the binomial theorem:   We have used the binomial theorem to expand (1-x) (ru)+ (s-v)+(t-w) .
To prove that each power basis element can be expressed as a linear combination of Bernstein polynomials, we use the degree elevation formulas and induction to calculate the following: where the induction hypothesis was used.

A Matrix Representation of Bernstein Polynomials for Triple Sequence Spaces
For example, until the 1990s as a systematic review of Bernstein polynomials theory (Lorentz, 1986;Videnskii and Polynomials, 1990) are also presented. New articles are constantly coming out (Esi and Catalbas, 2014) and new applications and generalizations are discovered (Li et al., 1997;Petrone, 1999). A generalization of Bernstein polynomials containing q-integers was proposed by (Lupas, 1987). However, the q-analogue of the Bernstein operator as handled by Lupa-s gives rational functions rather than polynomials.
In many applications, it is helpful to Bernstein polynomials of a matrix formulation. When looking at only a linear combination in terms of point products, they are simple to develop.
Given a polynomial that can be expressed as a linear combination of Bernstein's elementary functions: where the bijk are the coefficients of the power base used to determine the respective Bernstein polynomials.
Notice that in such a case the matrix is the lower triangular matrix.
In the quintic case (n = 5), the matrix representation is: