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In Ryszard Smarzewski's paper ``BEST REGULARIZED UNIFORM APPRO- XIMATIONS", he considers the subspace C(T) in the Hilbert space L2(T,.mu.), defines the new norm which is equivalent to the uni- form norm in C(T) and defines the best regularized uniform appro- ximations in M to a function x in C(T). Then he replaces the best uniform approximations by the best regularized uniform approxi- mations and concludes some results about best regularized uniform approximations. Finally, he presents a convergent regularized algorithm of the Remes-type for the approximate computation of best uniform approximations by elements of any finite dimensional subspace of C(T). In this paper, our main purpose is to extend the results in Smarzewski' s paper by considering C(T) as a subspace of the Banach space Lp(T,.mu.) with 1<p<.inf..
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