The functional-analytic properties of the limit \(q\)-Bernstein operator
Summary: The limit \(q\)-Bernstein operator \(B_q\), \(0 < q < 1\), emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in the \(q\)-boson theory to describe the energy distribution in a \(q\)-analogue of the coherent state. Lately, the limit \(q\)-Bernstein operator has been widely under scrutiny, and it has been shown that \(B_q\) is a positive shape-preserving linear operator on \(C[0, 1]\) with \(||B_q|| = 1\). Its approximation properties, probabilistic interpretation, eigenstructure, and impact on the smoothness of a function have been examined. In this paper, the functional-analytic properties of \(B_q\) are studied. Our main result states that there exists an infinite-dimensional subspace \(M\) of \(C[0, 1]\) such that the restriction \(B_q|_M\) is an isomorphic embedding. Also, we show that each such subspace \(M\) contains an isomorphic copy of the Banach space \(c_0\).
scientific article; zbMATH DE number 6134537
scientific article