publication . Article . 2007

local approximations based on orthogonal differential operators

Aleksandar Ignjatovic;
  • Published: 23 May 2007 Journal: Journal of Fourier Analysis and Applications, volume 13, pages 309-330 (issn: 1069-5869, eissn: 1531-5851, Copyright policy)
  • Publisher: Springer Science and Business Media LLC
Let M be a symmetric positive definite moment functional and let \(\{P_n^{\cal M}(\omega)\}_{n\in {\Bbb N}}\) be the family of orthonormal polynomials that corresponds to M. We introduce a family of linear differential operators \({\cal K}^n =(-i)^nP_n^{\cal M}(i\frac{d}{dt})\), called the chromatic derivatives associated with M, which are orthonormal with respect to a suitably defined scalar product. We consider a Taylor type expansion of an analytic function f(t), with the values f(n) (t0) of the derivatives replaced by the values \({\cal K}^n[f](t_0)\) of these orthonormal operators, and with monomials (t − t0)n/n! replaced by an orthonormal family of "specia...
free text keywords: Applied Mathematics, Analysis, General Mathematics, Continuous linear operator, Differential operator, Analytic function, Orthonormal basis, Taylor series, symbols.namesake, symbols, Bessel function, Positive-definite matrix, Orthogonal polynomials, Mathematical analysis, Combinatorics, Mathematics
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