Weighted Littlewood-Paley Theory and Exponential-Square IntegrabilitySpringer, 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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... t seem remarkable, that's only because we've been spoiled by a too-close familiarity with functional analysis. Now ... t seem like a big deal; but it is, precisely because of the sum's cancelation. Inequality 2.11 says that there isn't ...
... (t), a function of time, as if it were a signal. Then f2 gives one measure of f's average amplitude. The expression S(f)22, which is a sum of squares of f's component pieces, provides a measure of the energy of the signal. Equation 2.9 ...
... (t) dt. {t: |f(t)|>X/2} We plug in our estimate for |{x : Mg(f)(a) > X}| and apply 2.14: /(MAD) as of x- |: | no-yo" * d}\ 0 2|f(t) = / |f(t) |/ 2pxP-2 e dt 0 – or "- p–1 -2';*)//0/0." -2'; ...
... (t) C. < c//0 (/ £). < 1 + | |f(t) log" (f(t)) dt < C / |f(a) log(e.--|f(a)) day, I proving One direction of 2.15. To show the other direction in 2.15, define {I}}, (k = 0, 1, 2, ...) to be the family of maximal dyadic I! C I such that ...
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Cuprins
1 | |
9 | |
Exponential Square 39 | 38 |
Many Dimensions Smoothing | 69 |
The Calderón Reproducing Formula I | 85 |
The Calderón Reproducing Formula II | 101 |
The Calderón Reproducing Formula III | 129 |
Schrödinger Operators 145 | 144 |
Orlicz Spaces | 161 |
Goodbye to Goodλ | 189 |
A Fourier Multiplier Theorem | 197 |
VectorValued Inequalities | 203 |
Random Pointwise Errors | 213 |
References | 219 |
Index 223 | 222 |
Some Singular Integrals | 151 |
Alte ediții - Afișează-le pe toate
Weighted Littlewood-Paley Theory and Exponential-Square ..., Ediția 1924 Michael Wilson Previzualizare limitată - 2008 |