By Sergei Mihailovic Nikol’skii (auth.)
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Additional resources for Approximation of Functions of Several Variables and Imbedding Theorems
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.)