The quantum analogue of Bernstein operators \({\mathcal {B}}_{m,q}\) reproduce the linear polynomials which are therefore eigenfunctions corresponding to the eigenvalue \(1,~\forall ~ q>0\). 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.

中文翻译：

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q-Bernstein算子的本征结构

Bernstein算符\（{\ mathcal {B}} _ {m，q} \）的量子模拟可再现线性多项式，因此它们是对应于特征值\（1，〜\ forall〜q> 0 \）的特征函数。本文讨论了（i）\（1> q> 0 \）和（ii）\（q> 1 \）情况下q -Bernstein算子的其余本征结构以及本征函数零的独特行为。在MATLAB的帮助下，对一些本征函数及其根进行了图形分析。此外，讨论了q -Bernstein算子对角化的矩阵表示。