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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Sometimes this collection is a complete orthonormal family for L2, but it doesn't have to be. The square function S(f)(x) is defined as a weighted sum (or integral) of the squares of the inner products, |〈f,φ〉|2, where φ belongs to ...
The usefulness of the square function (in its many guises) comes chiefly from the fact that, for many linear operators T, S(T(f)), the square function of T(f), is bounded pointwise by a function ̃S(f), where ̃S(·) is an operator similar ...
The expert will see the close connections between wavelets and the material in chapters 5–7. The non-expert doesn't have to worry about them to understand the material; but, should he ever ...
Suppose it's a variable t. We typically want to show that there is a positive, finite constant C so that, for all t under consideration, A(t) ≤ CB(t). (1.2) We often want to prove such inequalities because they help us prove equations.
An example of such a pair of functions is A(t)=t(log(e+t)) and B(t)=t(log(534-H t”)), where the range of admissible t's is [0, oo); the reader should check this. A more interesting example is given by A(t)=t/(log(e.-- t))” and B(t) ...
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Goodbye to Goodλ
A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals