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On some quadrature rules with Laplace end corrections

Bogusław Bożek, Wiesław Solak, Zbigniew Szydełko (2012)

Open Mathematics

We investigate quadrature rules with Laplace end corrections that depend on a parameter β. Specific values of β yield sixth order rules. We apply our results to approximating the sum of slowly converging series s = Σi=1∞ f(i + 1/2) where f ∈ C 6 with its sixth derivative of constant sign on [m, ∞) and ∫ m∞ f(x)dx is known for m ∈ ℕ. Several examples show the efficiency of this method. This paper continues the results from [Solak W., Szydełko Z., Quadrature rules with Gregory-Laplace end corrections,...

On some shift invariant integral operators, univariate case

George A. Anastassiou, Heinz H. Gonska (1995)

Annales Polonici Mathematici

In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and...

On the Approximation by Convolution Operators in Homogeneous Banach Spaces of Periodic Functions

Draganov, Borislav R. (2011)

Mathematica Balkanica New Series

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.The paper is concerned with establishing direct estimates for convolution operators on homogeneous Banach spaces of periodic functions by means of appropriately defined Kfunctional. The differential operator in the K-functional is defined by means of strong limit and described explicitly in terms of its Fourier coefficients. The description is simple and independent of the homogeneous Banach space. Saturation of such...

On the approximation of real continuous functions by series of solutions of a single system of partial differential equations

Carsten Elsner (2006)

Colloquium Mathematicae

We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function f : ℝ s → ℝ can be approximated with arbitrary accuracy by an infinite sum ∑ r = 1 ∞ H r ( x ₁ , . . . , x s ) ∈ C ∞ ( ℝ s ) of analytic functions H r , each solving the same system of universal partial differential equations, namely P ( x σ ; H r , ∂ H r / ∂ x σ , . . . , ∂ ⁵ H r / ∂ x ⁵ σ ⁵ ) = 0 (σ = 1,..., s).

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