Approximating smooth functions using algebraic-trigonometric polynomials

The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form $p_n(t)+\tau_m(t)$, where $p_n(t)$ is an algebraic polynomial of degree $n$ and $\tau_m(t)=a_0+\sum_{k=1}^ma_k\cos k\pi t+b_k\sin k\pi t$ is a trigonometric polynomial of order $m$. The precise order of approximation by such polynomials in the classes $W^r_\infty(M)$ and an upper bound for similar approximations in the class $W^r_p(M)$ with $\frac43<p<4$ are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously.
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