Applications of Bernstein-Durrmeyer operators in estimating the covering number

Abstract

The paper deals with estimates of the covering number for some Mercer kernel Hilbert space with Bernstein-Durrmeyer operators. We first give estimates of l2 — norm of Mercer kernel matrices reproducing by the kernels $$ K^{\left( {\alpha ,\beta } \right)} \left( {x,y} \right): = \sum\limits_{k = 0}^\infty {C_k ^{\left( {\alpha ,\beta } \right)} Q_k ^{\left( {\alpha ,\beta } \right)} \left( x \right)Q_k ^{\left( {\alpha ,\beta } \right)} \left( y \right)} $$ , where Qk(α,β) (x) are the Jacobi polynomials of order k on (0,1),Ck(α,β) > 0 are real numbers, and from which give the lower and upper bounds of the covering number for some particular reproducing kernel Hilbert space reproduced by K(α,β)(x,y).