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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If x ∈ Rd—say, x = (x 1, x2, ... , xd)—then we define x's norm, x, to be ( ∑ d1x2n)1/2. This norm has the good property that, if y = (y1, y2, ..., yd) is any other vector in Rd, and |yn |≤|x n | for each n, then y ≤ x.
has the same domination property possessed by the Euclidean norm on Rd: if g = ∑ γnφn and |γn |≤|λ n | for all n, then g2 ≤ f2. Even better, if, for some ε > 0, we have |γn | ≤ε|λn | for all n, then g2 ≤ εf2.
Roughly speaking, this includes: the theory of the Lebesgue integral (in 1 and d dimensions), Lp spaces in Rd (1 ≤ p ≤ ∞) and their duals, and some functional analysis. We also assume that the reader knows a little about the Fourier ...
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 ...
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A Fourier Multiplier Theorem
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