Download e-book for kindle: Approximation Theory in the Central Limit Theorem: Exact by V. Paulauskas, A. Rackauskas

By V. Paulauskas, A. Rackauskas

ISBN-10: 9401177988

ISBN-13: 9789401177986

ISBN-10: 9401178003

ISBN-13: 9789401178006

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Additional info for Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces

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IHj [ ¢kCY1-J I h / B J =1 Ih i t = II f(x + y) I: J =k +1 B ::;. )-IHj [ thCY11 Ih Ih i t J sup If(x) I [IHJ[ thCY11 Ih B Ih i t J J =k +1 x ElI = 1111100. )_llh Ij 1 I k ). )-lgU)(O)(h)i1 = o( Ih Ik). (ylJ Ih Ij1(~), 1v'tJ Ih j? 1. 19). 0 For the measurable function f : B -+ R we shall denote iuf (x) = f f (x B + y)iu(~). 24. If the operator u ElI"7(H, B), then for each bounded measurable function f: B -+ R the function "lui E c:"(B, R), and for each k ? 1 it holds that sup sup I (iuf)£k)(x)(ht I ~ IIflloov'k!

21. 18) I, ... k)-unifonn, where the constant C depends only on the constants Co, C 1, ... (x):=f(x)4>(e- 1f(x». By denoting It I ~I, 4>(t) =0, if It I ::;1/2. t = e-1f(X), we have g~(x)(h) = f~(x)(h)(4>(t) + 4>'(t) . t), x E B, hE F, 34 CHAPTER 2 g~' (x)(h)2 = f~' (x)(h)2(#,t) + tt/J'(t»2 + f:-l(f~(x)(h»2(2tP'(/) + tt/J"(t». 19) we obtain Ig~(x)(h) IIx II ~ f:(2C o)-t, by the use of estimates I ~Cllhll, Ig~'(x)(h)21 ~C·e-lllhIl2. By induction on i ~ IIglli ~ eel - i, 1 we prove that i = 1, ...

2) c F such that E~lllxj liP < 00 the series E~leju(Xj) Proof The implication (b) => (a) is trivial. We shall prove (a) => (b). LetX b ... e. <;11', P'). e. v. ej, i = 1, ... 91', P xP'). By E we shall denote the mean with respect to the measure P x P' and, by E', to the measure P'. 4) IIXj(w)IIP. 2). We shall prove the implication (a) => (b). ,,(B). s. e. one can find sequences an ->00, x/, EF and a Rademacher sequence &5n), j = 1, ... , kn' n ~ 1, such that ~=la,;l < 00 and for all n ~ 1 IIj~l e5n)u (x5n» W~ an j~l IIx)n) liP.

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Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces by V. Paulauskas, A. Rackauskas

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