The Direct Theorem of the Theory of Approximation of Periodic Functions with Monotone Fourier Coefficients in Different Metrics

We study the problem of order optimality of an upper bound for the best approximation in L-q(T) in terms of the lth-order modulus of smoothness (the modulus of continuity for l = 1): En-1(f)(q) <= C(l, p, q) (Sigma(nu=n+1)nu(q sigma-1) omega(q)(l) (f; pi/nu)(p))(1/q), n is an element of N, on the class M-p(T) of all functions f is an element of L-p(T) whose Fourier coefficients satisfy the conditions a(0)(f) = 0, a(n)(f) down arrow 0, and b(n)(f) down arrow 0 (n up arrow infinity ), where l is an element of N, 1 < p < q < infinity, l > sigma = 1/p - 1/q, and T=(-pi, pi]. For l = 1 and p >= 1, the bound was first established by P. L.Ul'yanov in the proof of the inequality of different metrics for moduli of continuity; for l > 1 and p >= 1, the proof of the bound remains valid in view of the L-p-analog of the Jackson-Stechkin inequality. Below, we formulate the main results of the paper. A function f is an element of M-p(T) belongs to L-q(T), where 1 < p < q < infinity, if and only if Sigma(infinity)(n=1)n(q sigma-1) omega(q)(l) (f; pi/n)(p) < infinity, and the following order equalities hold: (a) En-1(f)(q)+n(sigma)omega(l)(f; pi/n)(p) asymptotic to (Sigma(infinity)(nu=n+1) nu(q sigma-1) omega(q)(l) (f; pi/v)(p))(1/q), n is an element of N, (b) n(-(l-sigma)) (Sigma(n)(nu=1) nu(p(l-sigma)-1) E-nu-1(P)(f)(q))(1/p) asymptotic to (Sigma(v=n+1) nu(q sigma-1) omega(q)(l) (f;pi/nu)(p))(1/q), n is an element of N. In the lower bound in equality (a), the second term n(sigma)omega(l)(f; pi/n)(p) generally cannot be omitted. However, if the sequence {omega(l)(f; pi/n)(p)}(n=1)(infinity) or the sequence {En-1(f)(p)}(n=1)(infinity) satisfies Bari's (B-l((p)))-condition, which is equivalent to Stechkin's (S-l)-condition, then En-1(f)(q) asymptotic to (Sigma(infinity)(nu=n+1) nu(q sigma-1) omega(q)(l)(f;pi/nu)(p))(1/q), n is an element of N. The upper bound in equality (b), which holds for every function f is an element of L-p(T) if the series converges, is a strengthened version of the direct theorem. The order equality (b) shows that the strengthened version is order-optimal on the whole class M-p(T).