Real and Complex Analysis

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Two Dimensional Signal And Image Processing

Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

Proofs from THE BOOK , by Martin Aigner and Günter M. Ziegler. Pp. 199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag).

The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and other hypersurfaces. As the field is quite large and has so many applications, it will be impossible to completely survey the field, but we will try to at least give the main ideas and developments in this area. 
The restriction problem are connected to many other conjectures, notably the Kakeya and Bochner-Riesz conjectures, as well as PDE conjectures such as the local smoothing conjecture. For reasons of space, we will not be able to discuss all these connections in detail; our main focus will be on proving restriction theorems for the Fourier transform.

Some Recent Progress on the Restriction Conjecture

Several school districts use assignment systems that give students an incentive to misrepresent their preferences. We find evidence consistent with strategic behavior in Cambridge. Such strategizing can complicate preference analysis. This paper develops empirical methods for studying random utility models in a new and large class of school choice mechanisms. We show that preferences are non-parametrically identified under either sufficient variation in choice environments or a preference shifter. We then develop a tractable estimation procedure and apply it to Cambridge. Estimates suggest that while 82% of students are assigned to their stated first choice, only 72% are assigned to their true first choice because students avoid ranking competitive schools. Assuming that students behave optimally, the Immediate Acceptance mechanism is preferred by the average student to the Deferred Acceptance mechanism by an equivalent of 0.08 miles. The estimated difference is smaller if beliefs are biased, and reversed if students report truthfully.

/pdf/demand-analysis-using-strategic-reports-an-application-to-a-2p2kw8ehds.pdf

Demand Analysis Using Strategic Reports: An Application to a School Choice Mechanism

On the Lp−Lq norm of the Hankel transform and related operators

We prove some restriction theorems for flat homogeneous surfaces of codimension greater than one.

/pdf/some-sharp-restriction-theorems-for-homogeneous-manifolds-52b15vygvc.pdf

Some sharp restriction theorems for homogeneous manifolds

We prove new Pitt inequalities for the Fourier transforms with radial and non-radial weights using weighted restriction inequalities for the Fourier transform on the sphere. We also prove new Riemann–Lebesgue estimates and versions of the uncertainty principle for the Fourier transform.

/pdf/pitt-inequalities-and-restriction-theorems-for-the-fourier-3sg4n8m07d.pdf

Pitt inequalities and restriction theorems for the Fourier transform

/pdf/interpolation-theory-and-applications-21n7rqqann.pdf

Interpolation Theory and Applications

My Academic Life (D Waterman) Reminiscences (L J Lardy & J L Troutman) On Concentrating Idempotents, A Survey (M Ash) Variants of a Selection Principle for Sequences of Regulated and Non-regulated Functions (V V Chystyakov et al) Local Lp Inequalities for Gegenbauer Polynomials (L De Carli) General Monotone Sequences and Convergence of Trigonometric Series (M Dyachenko & S Tikhonov) Using Integrals of Squares of Certain Real Valued Special Functions to Prove that the Polya Ξ*(z) Function, the Functions Kiz(a), a > 0, and Some Other Entire Functions Have Only Real Zeros (G Gasper) Functions whose Moments Form a Geometric Progression (M E H Ismail & X Li) Characterization of Scaling Functions in a Frame Multiresolution Analysis In H2G (K Kazarian & A San Antolin) An Abstract Coifman-Rochberg-Weiss Commutator Theorem (J Martin & M Milman) Convergence of Greedy Approximation with Regard to the Trigonometric System (V Temlyakov) Functions of Bounded -Variation (F Prus-Wisniowski)

Laura De Carli

Papers

On the Lp−Lq norm of the Hankel transform and related operators

Some sharp restriction theorems for homogeneous manifolds

Pitt inequalities and restriction theorems for the Fourier transform

Interpolation Theory and Applications

Topics in classical analysis and applications in honor of Daniel Waterman