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

By M.M. Rao

Provides formerly unpublished fabric at the fundumental pronciples and homes of Orlicz series and serve as areas. Examines the pattern course habit of stochastic procedures. offers functional functions in information and likelihood.

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

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Example 13. Consider the function M p ( - ) , p > 2, defined by u IP Mp(u) = log(e+ \u\)' This was shown to be an TV-function of class A2(oo), in Krasnoselskii and Rutickii ([1], p. 30), for p = 2. The same assertion is true for p > 2 as well. 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.

Here we include some extensions of (6) in the context of Orlicz spaces, indicating the availability of a class of interesting applications and research problems in this area. 3 Notes on Young functions and general measures 33 Proposition 13. Let $ be a Young function andL*(R) be the corresponding Orlicz space on the Lebesgue measure triple (R, £, //) as above. 7/L*(R) is closed under convolution, in the sense that /, g € I/*(R) implies f * g is defined as (6) and f * g G L^(R), £/&en i/iere exists a constant Q < K < oo such that \\f * 9\\* < K\\f\\*\\9\\*> /,0€L*(R).

For linearity let / 6 L® and c e R. Then we can find an integer no > 1 such that |c| < 2n° and so $(cu) < $(2n°u) < K n °$(u),u > u0. Hence r cfrl it <"" iI •*• \ Jf )iff«/-*' ^^ r*~i *—^^ • •/[|/l>«o] By the convexity of p$ (or $), it follows that for any fi 6 L*,z — 1,2, p$(/i + /2) < |[p*(2/i) + p*(2/2)] < oo, with c = 2 in the above. Thus L* is linear. If //(f2) = +00, then UQ = 0 so that the same argument is valid showing the linearity of L*. 14 /. Introduction and Background Material For the converse implication when // is diffuse, let QQ € S,0 < /^(fJo) < oo, be a set on which fj, is diffuse.

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