A fast converging sampling operator

Abstract

We construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a $K$-functional. The case of weighted approximation is also considered. The weights are positive and power-type with non-positive exponents at infinity. This sampling operator preserves every algebraic polynomial.

Keywords

• Sampling operator
• sampling series
• weighted approximation
• direct estimate
• weak converse estimate
• modulus of smoothness
• $K$-functional

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