Strong approximation by Fourier transforms and Fourier series in \(L^ \infty\)-norm (Q1904371)

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 \({\mathcal L}_p(R) := \{f \in L^p(R): d_\nu(f,p) = O(\nu^{-1/p})\) as \(\nu \to \infty\}\) is introduced. Sufficient conditions for \(f\) to belong to the class \({\mathcal L}_p(R)\) in the case \(2 \leq p <\infty\), and necessary conditions for \(f\) to belong to \({\mathcal L}_p(R)\) 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 Fejér mean of its Fourier series is studied.