Móricz Ferenc honlapja

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Cím:Strong approximation by Fourier transforms and Fourier series in $L\sp \infty$-norm.
Szerző:Dang Vu Giang; M\'oricz, Ferenc
Forrás:J. Approximation Theory 83, No.2, 157-174 (1995).
Nyelv:English
Absztrakt:The question of the strong approximation of functions $f$ belonging to the space $L^p(R)$, $1 < p < \infty$, by its Dirichlet integral $s_\nu (f,x) = {1\over \pi} \int_R f(x - t)\ {\sin \nu t\over t} dt $ is studied. With the help of quotient $$d_\nu (f,p):= \Biggl|\biggl\{ {1\over \nu} \int^\nu_0 |s_\mu(f,\cdot) - f(\cdot)|^p d\mu \biggr\}^{1/p} \Biggr|_\infty, \quad \nu \in R_+,$$ the class ${\cal L}_p(R) := \{f \in L^p(R): d_\nu(f,p) = O(\nu^{-1/p})$ as $\nu \to \infty\}$ is introduced. Sufficient conditions for $f$ to belong to the class ${\cal L}_p(R)$ in the case $2 \leq p <\infty$, and necessary conditions for $f$ to belong to ${\cal L}_p(R)$ in the case $1 < p \leq 2$ are established. In the case, when the Dirichlet integral is replaced by the Riesz means similar results are obtained. Besides the problem of strong approximation of a periodic function by the partial sum or Fej\'er mean of its Fourier series is studied.} \RV{D.K.Ugulawa (Tbilisi) 
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