On approximation of Stepanov’s almost periodic functions by means of Marcinkiewicz

We study some questions of approximation of Stepanov’s almostperiodic functions of partial Fourier sums and means of Marcinkiewicz, when the Fourier exponents of functions under consideration have a limit point in infinity.Let (𝑝 ≥ 1 denote the class of Stepanov’s almost-periodic functions, whose Fourier exponents take the following form: 0 = 0, -𝑛 = - 𝑛, lim 𝑛→∞ = ∞, 0) is a partial sum of Fourier series. Let Φ (𝑡) is an arbitrary real continuous even function such that 1)Φ (0) = 1; 2)Φ (𝑡) = 0 (|𝑡| ≤ ). We set (𝑓;'; 𝑥) = Σ︁ | 𝑚|≤ 𝐴𝑚Φ ( 𝑚)𝑒𝑖 𝑚𝑥. Let 𝑆𝑝(𝑅) stand for the space of bounded functions 𝑓(𝑥) ∈ (𝑝 ≥ 1) with the norm ‖𝑓(𝑥)‖𝑆𝑝 = sup -∞ 0. Theorem. If 𝑓(𝑥) ∈ 𝑆𝑝, where Fourier exponents have no limit points at a finite distance, i.e. → ∞, then the following bound is valid 𝑅(𝑓;',𝑎) ≤ + - 𝐸Λ(𝑓)𝑆𝑝, and 𝑓(𝑥) - 1 + 1 Σ︁𝑘=0 𝑆𝑘(𝑓; 𝑥)⃦⃦⃦⃦⃦𝑆𝑝 ≤ 𝑛 + 1 Σ︁𝑘=0 𝐸𝑘(𝑓)𝑆𝑝, where - constant and 𝐸Λ(𝑓)𝑆𝑝 = inf 𝑓(𝑥) - Σ︁ | 𝑚|≤Λ 𝑚𝑥⃦⃦⃦⃦⃦⃦𝑆𝑝.