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Extra info for Approximation of Functions of Several Variables and Imbedding Theorems

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4. Averaging of functions according to Sobolev so that, applying the generalized Minkowski inequality and taking account of the fact that ffJ has its support on cr, we obtain III. - Ilip ~ (6) ~ JffJ(v) II/(x - ev) - l(x)IIp dv sup II/(x - v) - l(x)IIp - 0 Ivl<8 (8 _ 0). In the case p = 00 property (5) is not satisfied. However, if we suppose that g = JR. }), then (6) may be written in the form Ills - llico ~ sup I/(x - v) - I(x)l - 0 rvl

5. Generalized functions from which it follows that (7) If ft is a Marcinkiewicz multiplier and is at the same time an infinitely differentiable function of polynomial growth, then for 1 E Lp and a sequence {til of infinitely differentiable finite functions for which (7) is satisfied, we will have (8) In the first term of (8) ftJis understood in the sense (4). In the second equation (8) the transfer of ft across the comma is legitimate, since ft is an infinitely differentiable function of polynomial growth.

3. (1 + x~)r/2 (1 + Ix\2)-r/2 (r > 0; i = 1, "'j n).