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In harmonic analysis, Littlewood–Paley theory is a theoretical framework used to extend certain results about ''L''2 functions to ''L''''p'' functions for 1<''p''<∞. It is typically used as a substitute for orthogonality arguments which only apply to ''L''''p'' functions when ''p''=2. One implementation involves studying a function by decomposing it in terms of functions with localized frequencies, and using the Littlewood–Paley ''g''-function to compare it with its Poisson integral. The 1-variable case was originated by and developed further by Polish mathematicians A. Zygmund and J. Marcinkiewicz in the 1930s using complex function theory . E. M. Stein later extended the theory to higher dimensions using real variable techniques. ==The dyadic decomposition of a function== Littlewood–Paley theory uses a decomposition of a function ''f'' into a sum of functions ''f''ρ with localized frequencies. There are several ways to construct such a decomposition; a typical method is as follows. If ''f'' is a function on R, and ρ is a measurable set with characteristic function χρ, then ''f''ρ is defined to be given by : where the "hat" is used to represent the Fourier transform. Informally, ''f''ρ is the piece of ''f'' whose frequencies lie in ρ. If Δ is a collection of measurable sets which (up to measure 0) are disjoint and have union the real line, then a well behaved function ''f'' can be written as a sum of functions ''f''ρ for ρ ∈Δ. When Δ consists of the sets of the form :. for ''k'' an integer, this gives a so-called "dyadic decomposition" of ''f'': Σρ ''f''ρ. There are many variations of this construction; for example, the characteristic function of a set used in the definition of ''f''ρ can be replaced by a smoother function. A key estimate of Littlewood–Paley theory is the Littlewood–Paley theorem, which bounds the size of the functions ''f''ρ in terms of the size of ''f''. There are many versions of this theorem corresponding to the different ways of decomposing ''f''. A typical estimate is to bound the ''L''''p'' norm of (Σρ |''f''ρ|2)1/2 by a multiple of the ''L''''p'' norm of ''f''. In higher dimensions it is possible to generalize this construction by replacing intervals with rectangles with sides parallel to the coordinate axes. Unfortunately these are rather special sets, which limits the applications to higher dimensions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Littlewood–Paley theory」の詳細全文を読む スポンサード リンク
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