Abstract
Let w(x)=eâxÎČxα, wÂŻ(x)=xw(x) and let {pm(w)}m, {pm(wÂŻ)}m be the corresponding sequences of orthonormal polynomials. Since the zeros of pm+1(w) interlace those of pm(wÂŻ), it makes sense to construct an interpolation process essentially based on the zeros of Q2m+1:=pm+1(w)pm(wÂŻ), which is called âExtended Lagrange Interpolationâ. In this paper the convergence of this interpolation process is studied in suitable weighted L1 spaces, in a general framework which completes the results given by the same authors in weighted Lup((0,+â)), 1â€pâ€â (see [31], [28]). As an application of the theoretical results, an extended product integration rule, based on the aforesaid Lagrange process, is proposed in order to compute integrals of the typeâ«0+âf(x)k(x,y)u(x)dx,u(x)=eâxÎČxÎł(1+x)λ,Îł>â1,λâR+, where the kernel k(x,y) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [23]. It is shown that the âcompound quadrature sequenceâ represents an efficient proposal for saving 1/3 of the evaluations of the function f, under unchanged speed of convergence.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.