Approximation of functions of two variables in the spaces <i>L</i><sub><i>р</i></sub>(ℝ<sup>2</sup>) and <i>L</i><sub><i>р</i></sub>(ℝ<sup>2</sup><sub>+</sub>) on directions
Let r ∈ ℕ, a,t ∈ ℝ, x = (x1, x2)∈ ℝ2, f : ℝ2 → ℂ,. Let's put r A(]a(f, x) = ^^(-1)r-kCkf (x1 + kt cos a, x2 + kt sin a). This paper is concerned with the problem, how the behavior of value j Ar,a(f, -)vn(t)dt at n - oc, where E С R, G € {R2, R+}, ipn € L1(E) -a positive kernel, the norm undertakes in space Lp(G) at 1 p oc, in space C(G) at p = oc is connected with structural properties of function f. We suppose that these structural properties are characterized by modules of a continuity on directions Ur,a(f,h)p,G = sup || Ar,a(f) ||p,G 0Let E and A be intervals in R+, such that A с E, f € Lp(G), a € [0, 2n], when G = R2, and a G [0, when G = R2. Let's put An,fc = / tkip„(t)dt. Then, if at some r G N and A lim AA"' r+1 = 0, limf ф„ = 0 all m € N next relations Am, r > 0, Am, r+1 take place and next equalities  An, E\A satisfied then relations J AIM, -)vn(t)dt E K, SUp t rwr,a(f,t)p,G are equivalented. Concrete methods of approximation are considered. In particular, it is established Corollary 1. Let p, G, a, f same as in theorem 1, II'( sin - \ 2 an,a{f,x) = - J а1М,х)(-^-) dt. Then relations ]han^00j^\\arha(f)\\piG K, sup i 1wi,q,(/,i)p>G equivalent.