Download Applications of Orlicz Spaces (Pure and Applied Mathematics) by M.M. Rao PDF

By M.M. Rao

ISBN-10: 0585425140

ISBN-13: 9780585425146

ISBN-10: 0824707303

ISBN-13: 9780824707309

Provides formerly unpublished fabric at the fundumental pronciples and houses of Orlicz series and serve as areas. Examines the pattern course habit of stochastic tactics. presents sensible purposes in information and likelihood.

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Extra resources for Applications of Orlicz Spaces (Pure and Applied Mathematics)

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Proof. It was already noted that f*g = g*f whenever it is defined, and by hypothesis, Z/*(R) is closed under convolution as multiplication, so that it is an algebra. Then the mapping / h-» Tf defined as Tf(g) — f*g on the spaces T : L*(R) -> #(L*(R)), is a homomorphism, since for any h € L*(R) one has and the left side is also (/ * g) * h = Tf*g(h), so that TfTg = Tf*g = Tg*f = TgTf, and /•->•/ (the complex conjugate) is an "involution". , Loomis [1], Lemma on p. 92). Moreover, ||T/|| = | \Tf \\sp in this case.

As another example, let $ be a continuous Young function that vanishes only at the origin and suppose that the complementary Young function ^ of $ has the same properties (by Proposition 4, $ is an TV-function), then it is shown that ||/||$ = ||/||($) iff / = 0 (cf. Rao [5], Lemma 1). Since for $3 of the above Young function, which is not an TV-function, this result is not applicable (it cannot hold since, in fact, the equality is valid for L 1 ). One of the important features of Orlicz spaces is that the two norms || • ||$ and || • ||($) are equivalent and rarely equal, in contrast to the Lebesgue spaces.

In fact, since Mp(0) = 0, and for u > 0 dMp _ pup~1(e + u) log(e + u) - up > ~d^(U} ~ (e + n)[log(e + u)] 2 ' 2 2 2 d Mp = puP- {[(e + u} log(e + u) - u} + g ( u ) } 2 (U) du (e + u) 2 [log(e + n)]3 where 2 1 g(u) = (p ~ 2}(e + w) 2 [log(e + n)] 2 - w2 + -u2 + -u2 log(e + u) > (p~ 2)u2 - u2 + —u2 + -n2 P P u2 P -- + - > 0. 1 N-functions and quantitative indices 11 Thus Mp(-) is convex increasing, and is an AT-function for p > 2 also. 00 M fu) ~ ^P' ^ 1S m ^2(00) as well. We shall see in Chapter 3 that the Orlicz sequence space ^(MP) and the Lebesgue sequence space lp have the same corresponding weakly convergent sequence coefficients.

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