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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.
List of contents
Some Assumptions.- An Elementary Introduction.- Exponential Square.- Many Dimensions; Smoothing.- The Calderón Reproducing Formula I.- The Calderón Reproducing Formula II.- The Calderón Reproducing Formula III.- Schrödinger Operators.- Some Singular Integrals.- Orlicz Spaces.- Goodbye to Good-?.- A Fourier Multiplier Theorem.- Vector-Valued Inequalities.- Random Pointwise Errors.
About the author
Michael Wilson received his PhD in mathematics from UCLA in 1981. After post-docs at the University of Chicago and the University of Wisconsin (Madison), he came to the University of Vermont, where he has been since 1986. He has held visiting positions at Rutgers University (New Brunswick) and the Universidad de Sevilla.
Summary
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.
Additional text
From the reviews:
"This engaging book by J. Michael Wilson concentrates on weighted inequalities of the forms … . The subject matter is presented in a fashion accessible to an advanced graduate student. Proofs of major … results are usually given in full. … There are a good number of exercises at the end of each chapter … . In addition there are many suggestions in the body of the text to prove or further investigate a given result." (Caroline P. Sweezy, Mathematical Reviews, Issue 2008 m)
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From the reviews:
"This engaging book by J. Michael Wilson concentrates on weighted inequalities of the forms ... . The subject matter is presented in a fashion accessible to an advanced graduate student. Proofs of major ... results are usually given in full. ... There are a good number of exercises at the end of each chapter ... . In addition there are many suggestions in the body of the text to prove or further investigate a given result." (Caroline P. Sweezy, Mathematical Reviews, Issue 2008 m)
