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) + Sigma(m)(k)= 1(a)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-infinity(r)(M) and an upper bound for similar approximations in the class W-p(r) (M) with 4/3 < p < 4 are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously.