Strong Approximation by Dirichlet Integrals in Lλ(R)-Norm, 1 < λ < ∞ Original Research Article

Let ƒ be a real valued function which belongs to Lr ≔ Lr(−∞, ∞) for some 1 ≤ r < ∞. We consider the cosine transform ƒ̂c, sine transform ƒ̂s, (complex) Fourier transform ƒ̂, and Hilbert transform ƒ̃ of ƒ. We study the strong approximation of order p, 0 < p < ∞, of ƒ and ƒ̃ by their Dirichlet integrals, respectively. We prove that the saturation class in Lλ-norm is the Lizorkin-Triebel space Fαλ, p, where α = 1/p, 2 ≤ p < ∞, and 1 < λ < ∞. To this effect, we introduce several so-called Littlewood-Paley functions and make use of a number of equivalence theorems. Our machinery is also appropriate to characterize the saturation class concerning the strong approximation of order p of a periodic function ƒ ∈ L12π ≔ L1(−π, π) by the partial sums of its Fourier series in Lλ2π-norm, where again 2 ≤ p < ∞ and 1 < λ < ∞.