ISSN 2348-1218 (print) International Journal of Interdisciplinary Research and Innovations ISSN 2348-1226 (online) Vol. 9, Issue 3, pp: (46-51), Month: July - September 2021, Available at: www.researchpublish.com
Application of Fractional Bernstein Polynomial Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong Province, China
Abstract: This paper uses the fractional Bernstein theorem to prove the fractional Weierstrass’s approximation theorem. And hence, we obtain that the set of fractional functions is dense in the set of continuous functions on a closed interval. The fractional Bernstein polynomial plays an important role in this article, which is the generalization of classical Bernstein polynomial. Keywords: fractional Bernstein theorem, fractional Weierstrass’s approximation theorem, fractional Bernstein polynomial.
I. INTRODUCTION Bernstein polynomial is the polynomial named after Russian mathematician Bernstein. It is a remarkable family of polynomials associated to any given function on the unit interval. If is continuous on , - , its -th Bernstein polynomial is defined by ( )
∑
. /. /
(
)
.
(1)
( ) converges uniformly to ( ) on , -, thus giving a constructive proof of the Bernstein showed in [1] that Weierstrass’s approximation theorem, which stated that any continuous function defined on closed interval can be approximated by polynomials. One might wonder why Bernstein created new polynomials for this purpose, instead of using polynomials that were already known to mathematics. Taylor polynomials are not appropriate; for even setting aside questions of convergence, they are applicable only to functions that are infinitely differentiable, and not to all continuous functions. On the other hand, the concept of fractional calculus originated from L’Hospital to Leibniz in 1695. One of the questions he asked was, “what will be the result about ⁄
calculus for the first time that
⁄ ⁄
√
⁄ ⁄
? ”.But until 1819, Lacroix gave the simplest result of fractional
. However, unlike the derivative of integer order, the fractional derivative is
not unique. There are some different definitions of fractional derivative: Riemann-Liouville (R-L), Grunwald-Letnikov, Caputo, Miller Ross sequence, and Jumarie type of modified R-L fractional derivatives. In recent decades, researchers have found that the fractional calculus operator has nonlocality property, that is, the next state of a system depends not only on its current state, but also on its historical state starting from the initial time. Therefore, it is very suitable to describe the material with memory and genetic properties in the real world. Compared with the integer order model, the fractional order model is perfect. Moreover, the analysis shows that the fractional order model is more practical than the integer order model. In addition, fractional calculus has wide applications in viscoelasticity, quantum mechanics, electromagnetism, electrochemistry, signal and image processing, vibration and oscillation, biology continuum mechanics, electrical engineering, fluid science, control theory, dynamics, finance, and so on [2-23]. For more details and applications of Jumarie’s modified R-L fractional derivative, we can see [24-39]. In this paper, we prove the fractional Bernstein theorem, and hence the fractional Weierstrass’s approximation theorem - (the set of all can be obtained. In other words, we show that the set of -fractional polynomials is dense in ,
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