Weierstrass quasi-interpolants

Abstract

In this paper, the expression of Weierstrass operators as differential operators on polynomials is used for the construction of associated quasi-interpolants. Then the convergence properties of these operators are studied.

Approximation by Baskakov quasi-interpolants

“…In general, both families of polynomial coefficients satisfy a recurrence relation. This has been proved [11] for Bernstein and Szász-Mirakyan operators and their associated Kantorovich and Durrmeyer versions, and also for Weierstrass operators [14].…”

Operator norms of Gauss-Weierstrass operators and their left quasi interpolants

“…Using the well-known estimate |H 2r (x)| ≤ 2 r (2r)!e x 2 /2 (see, e.g., [9, Subsection 1.5.1, p. 31]), Sablonnière [11,Theorem 6] proves, for r = 0, 1, 2, . .…”

“…as a differential operator on the space of algebraic polynomials [11,Theorem 1]. Here D denotes the differentiation operator.…”

“…For more smooth functions the operators W n possess the complete asymptotic expansion n with respect to the uniform norm in terms of a certain integral which cannot be exactly evaluated. He proved that r + √ 2 is an upper bound on this operator norm [11,Theorem 6].…”

“…, presented their explicit integral representations and derived a plenty of nice properties. In particular, Sablonnière[11, Theorem 5] expressed the operator norm of W…”