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Approximation of integrable functions by linear methods almost everywhere
T. V. Radoslavova V. A. Steklov Mathematical Institute, USSR Academy of Sciences
Abstract:
It is shown that
$2\pi$ periodic functions whose
$(r-1)$-th derivatives have bounded variation
$(r>0)$ can be approximated by de La Vallée-Poussin $\sigma_{n,m}(an\le m=m(n)\le An, 0<a<A<1)$ at almost all points with a rate
$o(n^{--r})$. For functions belonging to the class
$\operatorname{Lip}(\alpha,L)(0<\alpha<1)$, any natural
$N$, and a positive
$\varepsilon>0$, we have almost everywhere
$$
|f(x)-\sigma_{n,m}(f;x)|\le c(f,x)n^{-\alpha}\ln n\dots\ln_N^{1+\varepsilon}n,
$$
where $\ln_kx=\underbrace{\ln\dots\ln x}_k(k=1,2,\dots)$. For any triangular method of summation
$T$ with bounded coefficients we construct functions belonging to
$\operatorname{Lip}(\alpha,L)(0<\alpha<1)$ and such that almost everywhere,
$$
\varlimsup_{n\to\infty}|f(x)-\tau_n(f;x)|n^\alpha(\ln n\dots\ln_Nn)^{-\alpha}=\infty,
$$
where the
$\tau_n(f;x)$ are the means of the method
$T$.
UDC:
517.5
Received: 04.11.1974