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

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1 d)" ~ + n + 2, 0, cp) :;::::;- c u(lkl + n + 2, 0, cp). 1 1 and for IXil ~ 1, taking account of the fact that LIN is the portion of 1R. I,,+,k'+2 (u(n + Ikl + 2, 0, cp) + u(n + Ikl + 2, ei' cp)) dA. where ej is the unit vector directed along the Xj axis. p(k)(x) I ~ C1 k( u(n + Ikl + 2, 0, cp) + i~ u(n + Ikl + 2, ei' CP)), and we have proved inequality (4) for any k and l = 1. For an arbitrary l the proof is analogous; we need only integrate by parts in equation (7) 1 times instead of once. For cp, 1p E 5 put!

We will suppose that x, x' denote any elements of 9JL Obviously the elements y - Xn Wn=~-'-'d en + have unit norms, and for any o~ 1 - II(Xw n = 1 _ + fJwmll (_(X_ d+e n ~1uniformly for the (x, (x, = 1 fJ ~ 0 with -II(-(X + d+ en + _fJ_) Ily d+e m (_li_ + _fJ__) d+e n (X + fJ d +em fJ in question. = 1, d+fJ em ) y - xii x'il ~ d = 1')nm n,m-'>oo ~ 0, = 24 1. Preparatory information In such a case, from the definition of a uniformly convex space, IIWn - w",11 But O. CO) ---- II (X" X"') +1e" d +1) em d + en d + em Ili~ _d~II + 0 (1) =~d IIx" - xmll + 0(1) (n, m -+00), l d + en + em IIw,,-wmll = y (- - - - - - I , = d since the elements xn, xm are bounded in norm.

We do not intend to consider in all generality the case when the multiplicator is a product }'Il, where XELand Il is an infinitely differentiable function of polynomial growth. We will not need this in what follows. But there is one case which we shall need-the case of the factor V-lIl V, where V, p, eLand V moreover is a positive infinitely differentiable function of polynomial growth. If ! E L p , then the operation V-1IlVj = V-l(ll(Vj)) has meaning. Indeed, vj may be understood in the sense (1) or (6).

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