On Multiple Node Gaussian Quadrature Formulae

Abstract

Abstract. Let fip . -. , Hk be odd positive integers and »i = Z¡=.x(p¡ + 1). Let {«(.}|=j be an extended Tchebycheff system on [a, b]. Let L be a positive linear functional on U = span( {u,}). We prove that L has a unique representation in the

Worst-case optimal approximation with increasingly flat Gaussian kernels

“…-In Section 3 we present a generalisation, based on a theorem of Barrow [2], for optimal kernel quadrature rules [21, Chapter 5] that have both their points and weights selected so as to minimise the worst-case error. The result, Theorem 3.4, states that such rules, if unique, converge to the N -point Gaussian quadrature rule for the functional L, which is the unique quadrature rule Q(XG, wG) such that…”

Gaussian kernel quadrature at scaled Gauss–Hermite nodes

“…. , 2N −1 [3]. Appropriate control of the nodes of these quadrature rules would establish an exponential convergence result with the "double rate" M N = 2N.…”

O vosstanovlenii funktsionalov na ET-sistemakh

“…Obzor teorii gaussovyh kvadratur mo no na iti, naprimer, v [4]. V rabote [2] na ideny ocenki por dka toqnosti gaussovyh kvadratur dl polo itel nogo line inogo nepreryvnogo funkcionala na ET-sistemah, a tak e dokazany suwestvovanie i edinstvennost kvadratury, realizu we i dannu ocenku. Ocenki por dka toqnosti dl WT-sistem poluqeny v [1].…”

“…Pust w ζ ∈ C |m|+|µ|−1 [0, 1] -funkci , udovletvor wa sootnoxeni m (3), ET-sistema V |m|+|µ| vl ets w ζprodol eniem ET-sistemy U |m|+l , i µ i = m k = m (i = 1 : l, k = 1 : n).Esli suwestvuet kvadraturna formula(2), toqna dl L ζ na U |m|+l , to ona edinstvenna. .…”