Weighted Littlewood-Paley Theory and Exponential-Square Integrability
Springer, 31 dec. 2007 - 227 pagini
Littlewood-Paley theory is an essential tool of Fourier analysis, with applications and connections to PDEs, signal processing, and probability. It extends some of the benefits of orthogonality to situations where orthogonality doesn’t really make sense. It does so by letting us control certain oscillatory infinite series of functions in terms of infinite series of non-negative functions. Beginning in the 1980s, it was discovered that this control could be made much sharper than was previously suspected. The present book tries to give a gentle, well-motivated introduction to those discoveries, the methods behind them, their consequences, and some of their applications.
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Littlewood-Paley theory can be thought of as a profound generalization of the Pythagorean theorem. ... In analysis it is often convenient (and indispensable) to decompose functions f into infinite series, f(x) = ∑ λnφn (x), ...
... with its norm defined by f4≡ (∫ |f(x)|4dx ) 1/4. To make things specific, let's suppose that our functions are defined on [0,1). ... To each function f, one associates something called the square function of f, denoted S(f).
to the analysis of infinite series of non-negative functions, S(f)(x)=(∑|γi|2|ψi(x)|2)1/2; and that greatly simplifies things. We have already mentioned the practice, common in analysis, of cutting a function into infinitely many ...
where f ∗ g is the usual convolution, f∗g(x)= ∫ Rd f(x−y)g(y)dy= ∫ Rd f(y)g(x−y)dy defined for appropriate pairs of functions f and g. We use C∞ 0 (Rd) to denote the family of infinitely differentiable functions with compact ...
element (in the sense of set inclusion) of F that contains Q; such a maximal element must exist because of 1.1. ... For example, we might have two complicated but continuous functions f(t) and g(t), and want to show f(0) = g(0).
Ce spun oamenii - Scrie o recenzie
Exponential Square 39
Many Dimensions Smoothing
The Calderón Reproducing Formula I
The Calderón Reproducing Formula II
The Calderón Reproducing Formula III
Schrödinger Operators 145
Goodbye to Goodλ
A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals
Alte ediții - Afișează-le pe toate
Weighted Littlewood-Paley Theory and Exponential-Square ..., Ediția 1924
Previzualizare limitată - 2008