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Applications of Bernstein-Durrmeyer operators in estimating the covering number. (English) Zbl 1224.41074

Summary: The paper deals with estimates of the covering number for some Mercer kernel Hilbert spaces with Bernstein-Durrmeyer operators. We first give estimates of \(l^2\)-norm for Mercer kernel matrices reproduced by the kernels \[ K^{\alpha,\beta}(x,y):=\sum\limits^\infty_{k=0}C^{(\alpha,\beta)}_kQ^{(\alpha,\beta)}_k(x)Q^{(\alpha,\beta)}_k(y), \] where \(Q^{(\alpha,\beta)}_k(x)\) are the Jacobi polynomials of order \(k\) on \((0, 1)\), \(C^{(\alpha,\beta)}_k>0\) are real numbers. Using these estimates, we give the lower and upper bounds for the covering number of some particular reproducing kernel Hilbert spaces reproduced by \(K^{(\alpha,\beta)}(x,y)\).

MSC:

41A35 Approximation by operators (in particular, by integral operators)
41A25 Rate of convergence, degree of approximation
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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