Loukas Grafakos from University of Missouri-Columbia; February 27, 2020. Classical multiplier theorems and recent improvements; Host: Brett Wick
Abstract: A multiplier operator alters the frequency of input functions/signals via multiplication by a fixed function called a multiplier. Multiplier theorems provide sufficient conditions for multiplier operators to preserve integrability. The classical multiplier theorems of Marcinkiewicz and of Hörmander on Euclidean spaces will be reviewed and comparisons between different versions of these results (and examples) will be given. The main focus of the talk is to discuss recent optimal improvements of these theorems in terms of membership of the multipliers in appropriate scales of Sobolev classes.
Dominique Picard from University of Paris 7; October 12, 2017. Bayes adaptive procedures associated with operators; Host: Todd Kuffner.
Abstract: In lots of practical problems, the statistical issue is very much connected to a genuine geometry. It is obviously the case for data observed on geometrical objects (directional data, or data defined on some specific manifolds such as graphs, trees, or matrices), or data with a probabilistic structure involving an operator but also in the so-called linear inverse problems. In most cases the geometry is well described by a linear operator (the Laplacian for example).
The spectral decomposition of this operator plays an important role, generally translating a deep and intrinsic structure which leads to a natural choice of estimates (covariance kernel, spectral clustering, inverse problems). Moreover, the operator often induces a genuine regularization, leading to a regularity definition adapted to the structure of the data, which is fundamental in various situations: denoising, semi-supervised learning, classification...
We illustrate this problem by the example of bayesian functional estimation considering Gaussian processes as a-priori measures. In particular, the problem of adaptation shows the need for fitting the a priori distribution to an harmonic analysis of the structure of the data, and in particular we associate the choice of the Gaussian measure with the Laplacian of the structure. We also investigate the problem from the more explicit angle of an a priori measure on ‘manifold-wavelet’ coefficients.
We extend the results of Ghosal, Ghosh and van der Vaart , on the concentration a posteriori measures, for the case of geometrical data. We also relate this construction to a characterization of the regularity of Gaussian processes on geometric objects.
Anita Tabacco from the Polytechnic University of Torino, Italy ; May 14, 2014. Analytic and Geometric Features of Reproducing Groups; Host: Guido Weiss.
Abstract: We consider the (extended) metaplectic representation of the semidirect product G=Hd⋊Sp(d,R) between the Heisenberg group and the symplectic group. We introduce the notion of an admissible subgroup H of G. Under mild assumptions, admissibility of H is shown to be equivalent to the fact that the identity f=∫H(f,μe(h)φ)μe(h)φdh holds (weakly) for all f∈L2(Rd). We point out that the notion of admissibility captures natural geometric phenomena and show that it is a sufficient condition for a subgroup to be reproducing. The condition is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We prove in general that dimH≤d2+2d. Moreover If H⊂Sp(d,R), it is shown that dimH≤d2+1. Both bounds are proved to be optimal. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schroedinger representation of R2d or the wavelet representation of Rd⋊D, with D closed subgroup of GL(d,R). Finally, we shall provide new examples of reproducing groups of the type H=Σ⋊D, in dimension d=2
Maciek Paluszynski from Wroclaw University, Poland; Besov-Lipschitz Spaces associated with operators; September 27, 2012. Host: Guido Weiss.
Abstract: Original Besov-Lipschitz spaces were defined using the modulus of continuity. Later it was shown that they can be defined equivalently using the Gauss-Weierstrass kernel or the Poisson kernel. The new approach to these spaces uses a definition based on the heat semigroups generated by certain operators other than the Laplacian.
Steve Wainger from the University of Wisconsin-Madison; Recent progress on discrete singular Radon Transforms; April 25, 2012. Host: Guido Weiss.
Abstract: A well-studied operator defined on functions defined on the integers, ℤ, is Hf(m) = sum on non-zero integers of f (m-n k) / n, k odd. The techniques for finding properties of H involve the circle method of analytic number theory. We will briefly discuss this an indicate the critical role of the Fourier Transform on ℤ plays. We then turn to variants on discrete subgroups of nilpotent groups. The new results concern situations where the Fourier Transform does not seem to be a viable tool.
John Benedetto from the University of Maryland; Waveform design and balayage; March 3, 2011. Host: Guido Weiss.
Abstract: Constant Amplitude Zero Autocorrelation (CAZAC) sequences are a staple for waveform design of the type used in radar and communications theory.
Balayage is a concept that was introduced by Christoffel (1871) and developed by Poincare' and Brelot in potential theory. The theory of frames relates these ideas.
We shall construct new applicable CAZACs that depend on number theory and Fourier analysis. This is done in the context finite frames. We shall generalize Beurling's theorem on balayage in terms of parameterization over the space of Radon measures. The background requires the spectral synthesis ideas of Wiener and Beurling, and the results are formulated as sampling formulas. This is done in the context of Fourier frames.
Rodrigo Bañuelos from Purdue University; Lévy processes and Fourier multipliers; March 25, 2010. Hosts: Al Baernstein and Guido Weiss.
Abstract: Martingales arising from Brownian motion can be used to study properties of several classical Fourier multipliers, including the Hilbert transform, the Riesz transforms (and their Gaussian versions) and the Beurlin-Ahlfors operator. In this lecture we will explore similar techniques where stochastic integrals with respect to Brownian motion are replaced by similar quantitites arising from Lévy processes. This approach leads to a class of Fourier (Lévy) multipliers for which one gets Lp estimates that are similar to those obtained for the above mentioned singular integrals.
Elias Stein from Princeton University; Singular integrals and several complex variables: some new perspectives; May 8, 2008. Host: Al Baernstein.
Abstract: A survey of old and new ideas in the theory of singular integrals related to complex analysis, including some recent work with Nagel, Ricci, and Wainger.
The first Taibleson Lecture:
Hrvoje Šikić from the University of Zagreb (Croatia); Besov spaces on domains and Brownian motion; February 22, 2007. Hosts: Ed Wilson and Guido Weiss.
Abstract: Besov spaces form a large class of function spaces and they can be developed on the entire Euclidean space as well as on sub-domains. The development of Besov spaces started in 1959 and it took almost twenty years to include the full range of parameters. An important step was provided in 1964 by M.H.Taibleson, who applied the Hardy-Littlewood method to characterize Besov spaces on Euclidean spaces via the Poisson kernel and the Gauss-Weierstrass kernel. The research on Besov spaces on domains proved to be very challenging, as well, and it is active even today. From recent results on the potential theory of Brownian motion we were inspired to attempt the characterization of a large class of Besov spaces on domains via the kernel of the Brownian motion killed upon exiting the domain. Although we essentially revisited the original Taibleson's method, the proof ended up being demanding with several technical obstacles that do not appear in the original case. This is a joint work with M.H.Taibleson.