Approximation of functions in the mean by trigonometric polynomials

Approximation of the classes C β ψ H ω by generalized de la Valiée-Poussin sums

“…By using the last inequality, relation (7.66") in [4, p. 121], and inequality (9) for a(n) = g(n), we obtain the required estimate (14). Then there exists a constant K such that, for all n ~ N, the relation ~:(C~/-/to; V~'p h) < K~(n)o~(1/n) (15) is valid.…”

“…For ~(k) = k -r, r > 0, the classes C~Ho~ coincide with the well-known classes W~Ho~ whose approximating properties were studied in many works (see, e.g., [14][15][16][17][18][19]). A more detailed bibliography on these problems is given in works [4,20].…”

Approximation of the periodical functions of high smoothness by the rightangled Fourier sums

“…Нiкольського [1] випливає асимптотична рiвнiсть для то-чних верхнiх меж вiдхилень сум Фур'є на класах аналiтичних функцiй однiєї змiнної…”

Upper bounds of best one-sided approximations of the classes WrLΨ in the metric of L

“…(ii) that, for all f ~ WrV , we have On the other hand, it follows from [I, Eq. (9.25)] and from Nikol'skii's duality theorem [9] that, for any E > 0, a function ~, (x) exists such that H ~ lle(w) ~ I, ~ j~ const and "~ ~Zl ~D~n~ = n-Dr (nx) % (nx) dx § e, . and all 1~p~ oo, we have…”