On the relationship of a convolution type transformation and the best approximation of periodic functions

The space is understood as the collection of 2 periodic functions 𝑓(𝑥) ∈ 𝐿𝑝, for which |𝑓(𝑥)|𝑝 is Lebesgue integral with the norm ‖𝑓‖𝐿𝑝 = (︂∫︁ 2 0 |𝑓(𝑥)|𝑝𝑑𝑥 )︂1/𝑝 (1 ≤ 0 the condition ̂︀𝑦2(𝑥) = = 𝑥𝑙𝐹(𝑥), where 𝐹(𝑥) is the Fourier transform of some measure (𝑥), then the inequality 𝑊(𝑦2; 𝑓; ℎ)𝐿𝑝 ≤ 𝑀(𝑦1, 𝑦2, ) {︂∫︁ ℎ 0 (𝑦1; 𝑓; 𝑡)𝐿𝑝 𝑡 + + ℎ ∫︁ ∞ ℎ (𝑦1; 𝑓;𝐵𝑡)𝐿𝑝 𝑡 𝑙+1 }︂1 , where 𝑊(𝑦; 𝑓; ℎ)𝐿𝑝 = sup |𝑡|≤ℎ ‖𝐺(𝑓; 𝑦; ℎ; )‖𝐿𝑝 , = min(2, ) for 1 ≤