## 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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Littlewood-Paley theory can be thought of as a profound generalization of the Pythagorean theorem. ... In analysis it is often convenient (and indispensable) to decompose

**functions f**into infinite series, f(x) = ∑ λnφn (x), ...

... with its norm defined by f4≡ (∫ |f(x)|4dx ) 1/4. To make things specific, let's suppose that our functions are defined on [0,1). ... To each

**function f**, one associates something called the square

**function of**f, denoted S(f).

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 infinitely many ...

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 with compact ...

element (in the sense of set inclusion) of F that contains Q; such a maximal element must exist because of 1.1. ... For example, we might have two complicated but continuous

**functions f**(t) and g(t), and want to show f(0) = g(0).

### Ce spun oamenii - Scrie o recenzie

### 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 |