By Sergei Mihailovic Nikol’skii (auth.)
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Extra resources for Approximation of Functions of Several Variables and Imbedding Theorems
Thus we have proved that A is a Marcinkiewicz multiplier. 1. If the function u(x) = u(x1 , ... , then it also satisfies the conditions of this theorem if it is considered p,s a function of the k variables Xl, ••• , XI< (k < n), and accordingly it is a multiplicator relative to them. 5. Examples of Marcinkiewicz multipliers (in the· sense of Lp, 1 < p <(0). 1. sign x = .. n sign Xi' 1 2. (1 + Ixlll)-l (A> 0). 3. (1 + x~)r/2 (1 + Ix\2)-r/2 (r > 0; i = 1, "'j n).
But then, taking account of the fact that V-I is an infinitely differentiable function of polynomial growth, and V-IpVll ~ v-IpVl weakly. Therefore equation (12) may be obtained by a passage to the limit as l-7 00 of the already established equation (V-lp vl1, cp) = (Pll, cp) (cp E 5) . 1. General delinition 01 a multiplicator in Lp(1 ~ P soo). u(x) is a bounded function measurable on JR = JR n , so that ,U E 5'. We emphasize that if 1 E 5, then 1 E 5 is an infinitely differentiable function of polynomial growth.
Moreover, we will suppose that in (2) the series are extended only over k ;::;; 0, which does not affect the generality. Putting (5 ) c ei (l'x+,y) ". 5. 2 (13) it follows (nk = 2k) that: In the curly brackets on the right side of (7) there appears the function (see also (5)) (8) Tj,i,k,l VII';;I, where Vly;;1 is a coefficient not depending on x, y. It obviously may be considered as a segment of the Fourier series of the function (9) Accordingly, the quantity in curly brackets on the right side of (7) is a sum 2:2: of squares of segments of the Fourier series of the functions (9).
Approximation of Functions of Several Variables and Imbedding Theorems by Sergei Mihailovic Nikol’skii (auth.)