By Sergei Mihailovic Nikol’skii (auth.)

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Additional resources for Approximation of Functions of Several Variables and Imbedding Theorems

Sample text

Thus - IX) u' ~ y II = 1, 1. Because of the uniform convexity E(XiII) = u ~ y ,x~fl) = u':; Y), we get = u'. 7. It is frequently necessary for us to make use of the following fact, related to the theory of functions of a real variable. Suppose that I, Ik E Ip('8) (k = 1, 2, ... oo III - MLp(,ff) = 0 (k -+ 00), (1 ~ P :::;:00). Then there exists a subsequence {k,) of the natural numbers such that (2) lim Ik,(x) = I(x) for almost all x E '8. 3. (l = 1,2, ... ) are finite on ~' and equation (2) is satisfied for all tIC E~.

LI, Obviously (15) 8 n f- (kn In) _ Ck1 - 2S2 -,-. u ence u(x, y) - u 8 (x, y) = r 1 + r2 + ra. (;-, 1]) t(;, 1]) ei(xHYn) d; d17 r1 = (16) 2n LIN = E [k[,[II;£«N ~ 2n (kn In)' Ctlei~(kx+1Y) A. N i7(kIC+lu) 0 S S S S , N being a natural number, and s chosen in such a way that a nytural number: r2 = ~ 2n' f tX = !.... is :n; A. (;, 1]) i (;,1]) ei (xHIIr,) d;-, d1], JR,-LlN ra n };'}, (kn 8 , = --, -In) f~ (kn - , In) - e·i~(kIC+11l) 2S2 S S S S (k,l) Ikl III where the sum };' is extended over pairs such that either or is larger than IXN.

EUJtp(x) f _a(x) , from which we get (1) in view of (3). 2. Periodic functions from = sign co,,(x) n = L;. The functions sin(2"+1nx) (0;;;:;; x;;;:;; 1), 0, 1, ... , form an orthogonal and normal Rademacher system on [0, 1J. Here and in what follows we will frequently write A ~ B in place of A ;;;:;; cB, where c is a constant. For any double sequence {am"l of complex numbers and P > the inequalities ° I I (2' lam,,1 2)p/2 ~ J JI2' am"com(O) co,,(O')IP dO dO' ~ (2' la m,,1 2)p/2 (1) o 0 hold, with constants not depending on the am ...