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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... 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. In other words, the size of x, as measured by the norm ...
... defined by f4≡ (∫ |f(x)|4dx ) 1/4. To make things specific, let's suppose that our functions are defined on [0,1). The collection {exp(2πinx)}∞−∞ defines a complete orthonormal family in L2[0,1). Now, if f ∈ L4[0,1), then the ...
... defined on [0,1), there is a positive α such that exp(α(S(f))2) is integrable on [0,1)—and vice versa. (This is not quite like saying that |f| and S(f) are pointwise comparable, but in many applications they might as well be.) This ...
... definitions and conventions repeatedly. The definition of the Fourier transform we shall adopt is: ˆf(ξ) ≡ ∫ Rd f(x)e−2πix·ξ dx, originally defined for f∈ L1(Rd), and then by extension to f∈ L2. We have the Fourier inversion ...
... defined for appropriate pairs of functions f and g. We use C∞ 0 (Rd) to denote the family of infinitely differentiable functions with compact supports. The Schwartz class S(Rd) is the family of infinitely differentiable functions f ...
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 |