By Sergei Mihailovic Nikol’skii (auth.)
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Extra info for Approximation of Functions of Several Variables and Imbedding Theorems
Properties of the space Lp(t9') Then it follows from (1) and (2) that Ilfin> - f~")II},~ 1 (3 ) Ii where A = _Iifi + ft)'II}', n I 2 P in case (1) ) 2 and A = q in case (2). If now lI(Xfirll max (1 - + (1 - (X) f~'II)IJ) ~ 0, O~"'~l then But then the right side of (3) tends to zero, and the left side along with it. Hence Ilfi") - o. 6. Theorem. Suppose that E (in particular, Lp(~)) is a uniformly convex Banach space, m a subspace of E and y E E - m. Then there exists a unique element U E 9)1: yielding the best approximation to y among the elements of m: Ily - ull = inf Ily - xii.
Suppose given on 1R = 1R,. a lunction A(x) , having the lollowing properties. J. " exists and is continuous at any point x = (Xl' ... , XII) with i E ek, and it satislies the inequality Xi * 0, where Then A is a Marcinkiewicz multiplier. That is, there exists a constant "p not depending on M and I such that (4) lor aU IE LpThis theorem may be somewhat generalized in terms of generalized derivatives. We note that since A satisfies the property indicated in the theorem for k = 0, then it is bounded on 1R and continuous except at points belonging to the coordinate planes.
Banach . 4. Averaging of functions according to Sobolev for any Iyl < £5. Here ~" is the set of those x E ~ such that x for any y satisfying the inequality Iyi < £5. Theorem. Every function f(x) E Lp(~), 1 ~ P< large in Lp(~). 13. We shall use extensively also the following inequalities for 1~P~00: (1 ) 00 ( flak + bkl P)l1P ~ (f00 lakl P)l1P + (00 f Ibkl P) lip , 'f (2) 1 1 1 p+q=1, where ak and bk are arbitrary numbers, and q is taken to be 00 if P = 1. They are called respectively the Minkowski and Holder inequalities for sums.