## Weighted Littlewood-Paley Theory and Exponential-Square IntegrabilityLittlewood-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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The number AI (f) is known as f's

**Haar**coefficient for the interval I. By Bessel's Inequality, we immediately have: ... Elementary functional analysis now implies that, for all f e L*(R), the

**sum**XE, A1(f)h(r) converges to f in L*, ...

If we

**sum**up 2.6 over all dyadic I with length 2" and all dyadic J. with length 2", we get XL fiX1(a) – XL f/XJ (a) = XL ... To put it more generally: When can we use a vector-space decomposition of a function, via

**Haar**functions, ...

decomposition of a function, via Haar functions, to get information about the function's actual values—and vice-versa? ... than a sum of squares, but inequality 2.10 says that, on the average, this isn't true for

**sums of Haar**functions.

Now,

**Haar**functions aren't wavelets, strictly speaking, but they're near enough for this example. Suppose we have a function f which belongs to some Lp(R), with 1 < p < ∞. It is fundamentally important to know to what extent the

**sum**...

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### 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 - Afișați-le pe toate

Weighted Littlewood-Paley Theory and Exponential-Square ..., Ediția 1924 Michael Wilson,Professor Michael Wilson Previzualizare limitată - 2008 |