## 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

Rezultatele 1 - 5 din 89

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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 ...

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**defines**a complete orthonormal family in L2[0,1). Now, if f ∈ L4[0,1), then the coefficients λn ≡ ∫ 1 0 f(x) exp(−2πinx)dx are defined, and the infinite series, ∞∑ −∞ λn exp(2πinx), converges to f in the L4 sense, if we sum it ...

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**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 formula f(x) = ∫ Rd ˆf(ξ)e2πix·ξ dξ, which holds ...

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**Define**- (a) = f(a) if a £ UF, Q, QC ) = E.J., f dt if t e Q e FA. Then g is clearly bounded by 2"A almost everywhere. Set b = f – g. A little computation shows 0 a # UF, Q, b(a) = to- if a € Qe FA. We set b(Q)(a) = (f(a) – fo)xo(a) ...

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**define**fo = f – fl. Notice that |fo| < \/2 everywhere. Now we apply the previous splitting argument to fi, but use A/2 as our cut-off height, instead of A. We obtain two functions j and b. and a disjoint family of dyadic cubes J'A such ...

### 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 |