By Sergei Mihailovic Nikol’skii (auth.)

ISBN-10: 3642657117

ISBN-13: 9783642657115

ISBN-10: 3642657133

ISBN-13: 9783642657139

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

**Example text**

5) IIf. - flip ~ ° (13 ~ 0, II· lip = 11'IILp(lR)' 1 ~ P <(0). e. no matter how the open set g might be constructed, for each f E Lp(g) it is possible to find a family of infinitely differentiable functions f. e. the Sobolev averages, such that (5) is satisfied. (v) [f(x - 9? 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).

M o 0 = I lam"pl , 5 = I amnwm(O) w,,(O'}. Let A be a set of points (O,O') such that 15(0,0')1 > ~. CA the complement to A in Suppose S2 2 the unit square and IA I and ICA I their measures. A This means that 1 Accordin~ly, < ..!. + 2M If 4 1 IAI>-· or 8 1 1 o , 0 1 8 1 2P 151P dO dO' :2:: - . 5. Generalized functions which proves the first inequality (1) for p < 2. This proves inequalities (1) completely. Analogously one proves inequalities corresponding to (2) in the n-dimensional case: where the k = (k1' ...

Preparatory information with constants depending on IX but not on I and the distribution of the 8k, and with norms taken relative to the period. elvz = I 0,,(1) E L; " = ("1, ... , 'JI,,) ; k = (kl' ... , k,,)) , where and Here we are supposing that the e8 , s = 1, ... , n, can take on only the values ± 1. For such I and 11 we also have the inequalities (8) where the nonns are now taken relative to the n-dimensional period (O < xi < :at; i = 1, ... , n). o",(I)IP dX n e"o,,(I)IP dx" ~ Jdx,. e. (8), if we take into account the fact that Finally, for the functions (7) we may write the inequalities (9) with constants not depending on I.

### Approximation of Functions of Several Variables and Imbedding Theorems by Sergei Mihailovic Nikol’skii (auth.)

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