Conjugate functions and their connection with uniform rational and piecewise-polynomial approximations

As is well known, there is a close relationship between rational and piecewise-polynomial approximations of functions. This relationship is manifested in the most vivid way in the case of approximations in Lebesgue spaces $L_p$ for $0<p<\infty$, $1/p\notin\mathbb N$. In the present paper, in particular, it is shown that the rate of uniform rational approximation of functions is described rather well using the rate of uniform piecewise-polynomial approximations of the function itself and its conjugate function. The converse is also true.
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