Extensions of D. Jackson's Theorem on Best Complex Polynomial Mean Approximation1)
نویسندگان
چکیده
Sufficient conditions for the uniform convergence of polynomials Pniz) of best qth power approximation to a given function /(z) of a complex variable on a smooth curve Y were presented in an early paper by Jackson [1]. Although his theorem has found frequent application in approximation theory (see [2]), only slight extensions of the result are to be found in the literature. The present paper gives sufficient conditions for the convergence of the Pniz) to /(z) in the mean of order p>q on Y when convergence in the mean on Y of order p is known for some auxiliary sequence of polynomials, and so includes Jackson's theorem as the special case p = od. Further extensions ofthat theorem are given for best approximation by trigonometric polynomials, rational functions with prescribed and free poles, and bounded analytic and meromorphic functions. The method of proof uses standard best approximation arguments together with a recent result of Walsh [3] which we state as
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