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. |
Din interiorul cărții
Pagina vii
... 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 ...
... 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 ...
Pagina viii
... 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) ...
... 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) ...
Pagina ix
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. 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 ...
Pagina 2
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. 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 ...
Pagina 3
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 ...
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 ...
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 |
Termeni și expresii frecvente
absolutely convergent adapted functions argument assume belong bounded by(t Calderón formula Calderón reproducing formula Cauchy–Schwarz inequality chapter Chebyshev's inequality compact support compact-measurable exhaustion Corollary cubes Q define definition denote depending disjoint dt dy dyadic cubes dyadic doubling dyadic square function equation estimate everywhere exercise fe LP finishes the proof finite linear sums functions f Haar functions Hölder inequality holds implies Lebesgue Lebesgue measure Let f Let Q Littlewood-Paley theory locally integrable log(e–H LP norm LP spaces LP(w Ma(f Ma(v Martina Franca maximal function maximal operator measurable function measure space Moar non-negative Orlicz maximal Orlicz norms Orlicz space pointwise positive constant positive numbers Proof of Theorem Rademacher function radial Rd Rd reader result satisfies sequence subset sums of adapted sums of Haar support contained Suppose supremum Theorem 2.1 Tk(f yields Young function