## 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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**Haar function**expansions. We have put exercises at the end of almost every chapter. Some of them expand on topics treated in the text; some tie up loose ends in proofs; some are referred to later in the book. We encourage the reader ...

... function . Let D be the collection of dyadic intervals on R. For every I Є D , we let It and I , denote ...

**Haar functions**. We claim that { h ( 1 ) } IED is an orthonormal system for L2 ( R ) . We've just seen that [ h ( 1 ) ...

... Haar coefficient for the interval I. By Bessel's Inequality , we immediately have : I dx CMDRs / ) da for any ƒ € L2 ( R ) . We claim that the

**Haar functions**actually form a complete orthonormal system for L2 ( R ) . To see this , take ...

... LP spaces ? ( We'll explain what weighted spaces are in a bit . ) To put it more generally : When can we use a vector - space decomposition of a function, via

**Haar functions**, to get information 12 2 An Elementary Introduction.

Michael Wilson. decomposition of a function, via

**Haar functions**, to get information about the function's actual values—and vice-versa? This is one way to phrase the problem that Littlewood-Paley theory tries to address. Let's get back to ...

### Cuprins

1 | |

9 | |

Exponential Square 39 | 40 |

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 GoodX | 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 |