On eigenstructure of q -Bernstein operators
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On eigenstructure of q-Bernstein operators Ambreen Naaz1 · M. Mursaleen1,2 Received: 24 July 2020 / Revised: 18 September 2020 / Accepted: 10 November 2020 © Springer Nature Switzerland AG 2020
Abstract The quantum analogue of Bernstein operators Bm,q reproduce the linear polynomials which are therefore eigenfunctions corresponding to the eigenvalue 1, ∀ q > 0. In this article the rest of eigenstructure of q-Bernstein operators and the distinct behaviour of zeros of eigenfunctions for cases (i) 1 > q > 0, and (ii) q > 1 are discussed. Graphical analysis for some eigenfunctions and their roots are presented with the help of MATLAB. Also, matrix representation for diagonalisation of q-Bernstein operators is discussed. Keywords Bernstein operator · Upper-triangular matrix · Eigenvalues · Total positivity · q-Calculus Mathematics Subject Classification 65F15 · 65F08 · 41A36 · 15A18 · 41A10 · 39B42 · 05C62
1 Introduction In 1912, Bernstein [3] proposed the constructive proof of the famous Weierstrass [23] approximation theorem based on probabilistic method. For f ∈ C[0, 1], the Bernstein operators are defined as follows: Bm ( f ; x) =
m m k=0
B
k
x (1 − x) k
m−k
f
k , m
k, m ∈ N ∪ {0}.
(1.1)
M. Mursaleen [email protected] Ambreen Naaz [email protected]
1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2
Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan 0123456789().: V,-vol
6
Page 2 of 14
A. Naaz, M. Mursaleen
The eigen-structure of Bernstein operators are well described by Cooper and Waldron in [5]. There are various applications of eigenvalues and eigenvectors in different areas of technology, e.g., communication system, image compressing, designing bridges, exploring land for oil, electrical and mechanical engineering. In 1987, Lupa¸s [8] introduced a quantum-analogue of Bernstein operators (rational) and studied about approximating and shape-preserving properties. In 1997, Phillips [18] introduced another quantum-analogue of Bernstein operators (polynomials). For recent development on Bernstein operators, one can see [11–16] and [17]. In this paper we generalize the spectrum of Bernstein operators to provide appropriate tools for studying various problems of neural network [9,20], tomography [6], image compressing [2,21] and many more areas. This paper is organized as follows. In Sect. 2, we collect some of the notation and basic facts which will be used subsequently. Section 3 is devoted to the description of eigenvalues and eigenfunctions of Bm ( f , q; x) for the case q > 0. The zeros of eigenfunctions and the spectral properties of totally positive kernels are considered in Sect. 4. In Sect. 5, the eigenfunctions of q-Bernstein operators Bmq for some finite values of m have been calculated and represented graphically. Furthermore, from graphs one can see the motion of zeros of eigenfunctions for different values of q and from matrix representation, the diagonalisation of q-Ber
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