Strong Approximation by Fourier Transforms and Fourier Series in L∞-Norm Original Research Article

Let f be a complex-valued function belonging to Lp(R) for some 1 < p < ∞. We study the strong approximation of f, in L∞(R)-norm, by its Dirichlet integral, which is closely related to the Fourier transform of f. We prove sufficient conditions for f to belong to the saturation class Sp(R) in the case 2 ≤ p < ∞, and necessary conditions for f to belong to Sp(R) in the case 1 < p ≤ 2. As a consequence, we obtain a characterization of S2(R). We formulate a conjecture on the characterization of Sp(R) in the case 1 < p < 2, which is supported by our results on the strong approximation by Riesz means. Our machinery is also appropriate to prove sufficient or/and necessary conditions for the saturation class in connection with the strong approximation of a periodic function by the partial sum or Fejér mean of its Fourier series.