By Sergei Mihailovic Nikol’skii (auth.)

ISBN-10: 3642657117

ISBN-13: 9783642657115

ISBN-10: 3642657133

ISBN-13: 9783642657139

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**Sample text**

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. 5. J IP <{ Q JIlfll~ dO = IIfll~. J The transition from the 2nd to the 3rd and from the 6 th to the 7th terms are carried out on the basis of inequality (8) with ek = Wk(O), and those from the 4th to the 5th and thereupon to the 6th terms on the basis of inequality (3).

8. Lemma. 7. 7. the following lemma follows easily. 9. Lemma. Suppose that ~ = ~1 X ~2 is defined as in the preceding lemma, and suppose that lor the sequence offunctionslk E Lp(~) (k = 1,2, ... ) and the lunction I E Lp(~) the equation (1 ) is satislied. lim III - ftliL,,(8) =° (1:::;;: p ~oo). 26 Preparatory information 1. e. the functions f(x) and f*(x) are equivalent on (g. Proof. 8. (1) holds for all y E ~~. We may suppose that equation (2) also holds for almost all y E ~~. Thus, if Y E ~{, (1) and (2) hold simultaneously for it.

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).

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

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