## 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 52

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**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 coefficients λn ≡ ∫ 1 0 f(x) exp(−2πinx)dx are defined, and the ...

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**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 ...

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**suppose**there are two positive, finite constants, C1 and C2, such that, for all relevant t, C1 B(t) < A(t) < C2 B(t). (1.3) The inequalities 1.3 say that A and B are roughly the same size. An example of such a pair of functions is A(t)= ...

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**Suppose**I and J belong to D and I # J. If I O J = 0 it is trivial that /* h(J)(a) da: = 0.

**Suppose**IO.J # (). Without loss of generality, we may assume that I C J. But then, since I # J, the support of h(r) is entirely contained in J or ...

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**suppose**that (f h(t) = 0 for all I e D. The claim will be proved if we can show that f = 0. For this it is sufficient to prove that f is (a.e.) constant on (–oo,0) and (0, oo). A little computation shows that, for any I € D, Ar(f)=(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 |