Quadrature formulae with free nodes for periodic functions

Abstract

The problem of existence and uniqueness of a quadrature formula with maximal trignonometric degree of precision for 2π-periodic functions with fixed number of free nodes of fixed different multiplicities at each node is considered. Our approach is based on some properties of the topological degree of a mapping with respect to an open bounded set and a given point. The explicit expression for the quadrature formulae with maximal trignometric degree of precision in the 2-periodic case of multiplicities is obtained. An error analysis for the quadrature with maximal trigonometric degree of precision is given. Classification (1991): 41A55, 65D30 Mathematics Subject

Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness

“…By the assumption of uniqueness we conclude that x(ξ, η) = x, i.e., Proof The proof uses a modification of ideas of Bojanov [1], Shi [15], Shi and Xu [16] (for algebraic σ -orthogonal polynomials) and Dryanov [3]. For exactness a problem ϕ k (x, a) = 0, k = 1, .…”

“…The problem of existence and uniqueness of a quadrature formula of the maximal trigonometric degree of exactness with a fixed number of free nodes of fixed different multiplicities at each of nodes was considered by Dryanov [3], but only for the weight function w(x) = 1.…”

On a discrete variant of Bernstein’s polynomial inequality

“…Lemma 1 is a well-known quadrature formula (see for example [1] as a ready reference). Simple computations show that the quadrature is valid for t(θ) := e ikθ , 0 ≤ k ≤ 4n − 1.…”

Generalisane kvadraturne formule Gauss-ovog tipa

“…Dryanov je u radu [12] generalisao kvadraturne formule (3.14) na taj načinšto je posmatrao kvadraturne formule u kojimačvorovi imaju različite višestrukosti. Zapravo, posmatrao je kvadraturne formule sledećeg oblika Kvadraturne formule (3.14) i (3.15) generališemo takošto umesto težinske funkcije w(x) = 1 posmatramo proizvoljnu nenegativnu integrabilnu težinsku funkciju na intervalu [−π, π), koja je jednaka nuli samo na skupu mere nula.…”

“…, 2s ν . Jedan način za računanje težina je da se fundamentalni polinomi trigonometrijske Hermite-ove interpolacije (videti [13] i [12]) pomnože sa w i to integrali na intervalu [−π, π). Medutim konstrukcija Hermite-ovog interpolacionog trigonometrijskog polinoma je znatno komplikovanija od konstrukcije algebarskog Hermite-ovog interpolacionog polinoma.…”