Local Approximations Based on Orthogonal Differential Operators
نویسنده
چکیده
ABSTRACT. Let M be a symmetric positive definite moment functional and let {PM n (ω)}n∈N be the family of orthonormal polynomials that corresponds to M. We introduce a family of linear differential operators K = (−i) P n (i 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 (t0) of the derivatives replaced by the values K [f ](t0) of these orthonormal operators, and with monomials (t − t0) /n! replaced by an orthonormal family of “special functions” of the form (−1)K[m](t − t0), where m(t) =
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