On approximation of the functions of two variables by some fourier integrals

This paper we studies some issues on the deviation of the functions of two variables 𝑓(𝑥, 𝑦) defined on the whole two-dimensional space from integral mean values of their Fourier transforms in the metric of the space 𝐿𝑝(𝑅2) (1 ≤ 0 we consider 𝑆, (𝑓; 𝑥, 𝑦) = ∫︁ - ∫︁ - 𝐹(𝑡, 𝑧) exp(𝑖(𝑡𝑥 + 𝑧𝑦))𝑑𝑡𝑑𝑧 = = ∫︁ 0 ⎩ ∫︁𝑢 -𝑢 𝐴(𝑡, 𝑢)𝑑𝑡 + ∫︁𝑢 -𝑢 𝐴(𝑡,-𝑢)𝑑𝑡 + ∫︁𝑢 -𝑢 𝐴(𝑢, 𝑧)𝑑𝑧 + ∫︁𝑢 -𝑢 𝐴(-𝑢, 𝑧)𝑑𝑧 ⎭𝑑𝑢 = = ∫︁ 0 𝑆* 𝑢,𝑢(𝑓; 𝑥, 𝑦)𝑑𝑢, where 𝐴(𝑡, 𝑧) = 𝐹(𝑡, 𝑧) exp(𝑖(𝑡𝑥 + 𝑧𝑦)). This paper estimates the value 𝑅,𝑟(𝑓)𝐿𝑝 = ‖𝑓(𝑥, 𝑦) - 𝑈,𝑟(𝑓; 𝑥, 𝑦)‖𝐿𝑝, where 𝑈,𝑟(𝑓; 𝑥, 𝑦) = ∫︁ 0 (︂ 1 - 𝑟 )︂ 𝑆* 𝑢,𝑢(𝑓; 𝑥, 𝑦)𝑑𝑢. Theorem 1. If 𝑓(𝑥, 𝑦) ∈ 𝐿𝑝(𝑅2) (1 function show_eabstract() { $('#eabstract1').hide(); $('#eabstract2').show(); $('#eabstract_expand').hide(); }