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Polynomial Approximation of Piecewise Analytic Functions
Author(s) -
Saff E. B.,
Totik V.
Publication year - 1989
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-39.3.487
Subject(s) - mathematics , piecewise , polynomial , rate of convergence , constant function , uniform norm , norm (philosophy) , exponential function , complex plane , function (biology) , sign (mathematics) , analytic function , mathematical analysis , channel (broadcasting) , electrical engineering , evolutionary biology , political science , law , engineering , biology
For a function f that is piecewise analytic on [−1,1], we construct a sequence of polynomial approximants that converges to f at an exponential rate at each point of analyticity of f For the uniform norm on [−1,1], these polynomials approximate f to within a constant times the least possible error while, locally, the approximants give a ‘best possible’ rate of convergence. Moreover, unlike the best uniform approximants, the polynomials that we construct overconverge to an analytic continuation of f Also, we prove a conjecture of Grothmann and Saff concerning the rate of polynomial approximation in a region of the plane to a complex extension of the absolute value function. As the starting point for our proofs, we obtain ‘best possible’ polynomial approximations to the sign function.