## 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

...

**of f**against some other

**functions**ψn . If we are interested about convergence in the sense

**of**L2 (or “mean-square”), and if the φn's comprise a complete orthonormal family, then each ψn can be taken to be ̄φn, the complex conjugate

**of**...

...

**f**∈ L4[0,1), then the coefficients λn ≡ ∫ 1 0

**f**(x) exp(−2πinx)dx are defined, and the infinite series, ∞∑ −∞ λn ...

**functions**such that |〈g,φ〉| ≤ |〈

**f**,φ〉| for all φ, then S(g)(x) ≤ S(

**f**)(x) everywhere. The square

**function**S(

**f**) ...

Michael Wilson. 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 ...

Michael Wilson. 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 ...

Michael Wilson. element (in the sense of set inclusion) of F that contains Q; such a maximal element must exist ...

**functions f**(t) and g(t), and want to show f(0) = g(0). This is an immediate consequence of: |f(t) − g(t)| ≤ CB(t) limt ...

### 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 |