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.
... of f against some other functions ψn . If we are interested about convergence in the sense of L2 (or “mean-square”), and if the φn's comprise a complete orthonormal family, then each ψn can be taken to be ̄φn, the complex conjugate of ...
... f ∈ L4[0,1), then the coefficients λn ≡ ∫ 1 0 f(x) exp(−2πinx)dx are defined, and the infinite series, ∞∑ −∞ λn ... functions such that |〈g,φ〉| ≤ |〈f,φ〉| for all φ, then S(g)(x) ≤ S(f)(x) everywhere. The square function S(f) ...
Michael Wilson. 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 ...
Michael Wilson. 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 ...
Michael Wilson. element (in the sense of set inclusion) of F that contains Q; such a maximal element must exist ... functions f(t) and g(t), and want to show f(0) = g(0). This is an immediate consequence of: |f(t) − g(t)| ≤ CB(t) limt ...
Goodbye to Goodλ
A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals