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 <i>r</i> ∈<i> </i>ℕ, <i>a,t </i>∈<i> </i>ℝ, <i>x</i> = (<i>x</i><sub><i></i>1</sub>, <i>x</i><sub><i></i>2</sub>)∈<i> </i>ℝ<sup>2</sup>, <i>f </i>: ℝ<sup>2</sup> → ℂ,. Let's put r A(<sub>]a</sub>(f, x) = ^^(-1)<sup>r-k</sup>C<sup>k</sup>f (x<sub>1</sub> + <i>kt </i>cos a, x<sub>2</sub> + <i>kt </i>sin a). This paper is concerned with the problem, how the behavior of value <i>j </i>A<sup>r</sup>,a(f, -)vn(t)dt at <i>n </i>- oc, where E С R, <i>G € </i>{R<sup>2</sup>, R+}, <i>ip</i><sub><i>n</i></sub><i> € L</i><sub><i>1</i></sub><i>(E) </i>-a positive kernel, the norm undertakes in space <i>L</i><sub><i>p</i></sub><i>(G) </i>at 1 < <i>p < </i>oc, in space <i>C</i>(G) at <i>p </i>= oc is connected with structural properties of function f. We suppose that these structural properties are characterized by modules of a continuity on directions <i>Ur,</i><sub><i>a</i></sub><i>(f,</i><sup><i>h</i></sup><i>)p,G </i>= sup || A<sup>r</sup>,a(f) ||p,G 0<t<h Here is one of the results which had been received. Theorem 1. <i>Let E and A be intervals in R+, such that A </i>с <i>E, f </i>€ <i>L</i><sub><i>p</i></sub><i>(G), a </i>€ [0, 2n], <i>when G </i>= R<sup>2</sup>, <i>and a </i>G [0, <i>when G </i>= R<sup>2</sup>. <i>Let's put </i>A<sub>n</sub>,fc = / <i>t</i><sup><i>k</i></sup><i>ip„(t)dt. Then, if at some r </i>G N <i>and A </i>lim <sup>A</sup><sub>A</sub>"' <sup>r+1</sup> = 0, lim<i>f ф„ = 0 all m </i>€ N <i>next relations </i>A<sub>m</sub>, <sub>r</sub> > 0, A<sub>m</sub>, <sub>r</sub>+<sub>1</sub> < oc <i>take place and next equalities </i> An, E\A <i>satisfied then relations </i><sup><i>J AIM, </i>-)vn(t)dt </sup>E < <i>K, </i>SUp t <sup>r</sup>wr,a(f,t)<sub>p</sub>,G < k 46(0, те) <i>are equivalented. </i>Concrete methods of approximation are considered. In particular, it is established Corollary 1. <i>Let p, G, a, f same as in theorem 1, II'( </i>sin - \ <sup>2 </sup><i>a</i><sub><i>n</i></sub><i>,</i><sub><i>a</i></sub><i>{f,x) = - J а1М,х)(-^-) dt. Then relations ]ha</i><sub><i>n</i></sub><i>^</i><sub><i>00</i></sub><i>j^\\a</i><sub><i>rha</i></sub><i>(f)\\p</i><sub><i>i</i></sub><i>G </i>< <i>K, </i>sup i <sup>1</sup>wi,<sub>q</sub>,(/,i)<sub>p></sub>G < a" are <i>equivalent.</i>