Course unit organization. Prerequisites: Real Analysis. Some knowledge of Complex Analysis in one variable could be useful. Target skills and knowledge: The main task of this class will be to present and discuss the Restriction Problem for the Fourier Transform, one of the deepest open problems in Mathematical Analysis.
To accomplish this task we will start from the very definition of the Fourier transform in the n-dimensional Euclidean space E. Examination methods: Oral exam Assessment criteria: Check of the learning of the taught notions and on the ability of their application. This problem was solved in the early 20th century. The Hausdorff—Young inequalities, that we will prove and discuss, give all the inequalities of this form. In the sixties Elias M.
Stein discovered several interesting things about this question. One important discovery is that the curvature of the surface S matters. The unit sphere in the Euclidean space obeys inequalities that are qualitative different from the inequalities that obeys a unit flat disk of the same dimension.
When the surface S is the unit sphere, Stein made a conjecture about all the restriction estimates associated to the sphere and proved some nontrivial cases. The two dimensional case in this conjecture was then proved by Charles Fefferman.
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We will prove the theorems of Stein and Fefferman and then state and discuss the conjecture. In dimension three and higher, this conjecture is wide open and looks very difficult.
Planned learning activities and teaching methods: Lectures. Bestselling Series. Harry Potter. Popular Features. New Releases.go here
Math 247A (Harmonic Analysis).
Product details Format Hardback pages Dimensions x x Other books in this series. Convex Analysis Ralph Tyrell Rockafellar. Add to basket. Global Analysis Donald Clayton Spencer. Problems in Analysis Robert C.
Math A (Harmonic Analysis)
Back cover copy This book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, L superscript p estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group.
Review quote Elias M. About Elias M.
Stein Elias M. Stein is Professor of Mathematics at Princeton University. Rating details. Book ratings by Goodreads. Goodreads is the world's largest site for readers with over 50 million reviews.
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