Radon Transforms, Geometry, and Wavelets: Ams Special by Gestur Olafsson, Eric L. Grinberg, David Larson, Palle E. T.

By Gestur Olafsson, Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, Peter R. Massopust

This quantity relies on precise periods held on the AMS Annual assembly in New Orleans in January 2007, and a satellite tv for pc workshop held in Baton Rouge on January 4-5, 2007. It involves invited expositions that jointly symbolize a wide spectrum of fields, stressing striking interactions and connections among components which are mostly regarded as disparate. the most themes are geometry and indispensable transforms. at the one facet are harmonic research, symmetric areas, illustration conception (the teams contain non-stop and discrete, finite and endless, compact and non-compact), operator idea, PDE, and mathematical likelihood. relocating within the utilized course we stumble upon wavelets, fractals, and engineering subject matters similar to frames and sign and snapshot processing. the themes lined during this publication shape a unified entire, they usually stand on the crossroads of natural and utilized arithmetic. The articles conceal a extensive variety in harmonic research, with the most topics relating to quintessential geometry, the Radon rework, wavelets and body theory.These subject matters can loosely be grouped jointly as follows: body thought and purposes Harmonic research and serve as areas Harmonic research and quantity thought critical Geometry and Radon Transforms Multiresolution research, Wavelets, and functions

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Extra info for Radon Transforms, Geometry, and Wavelets: Ams Special Session January 7-8, 2007, New Orleans, Louisiana Workshop January 4-5, 2007 Baton Rouge, Louisiana

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6) dsX= SdsB, where B is BM(V). Calculations concerning X generally begin with an Ito analysis of the semimartingale f(X) for some suitable smooth real-valued function f. This is closely related to the martingale characterization (Stroock and Varadhan, 1979). It is actually more efficient first to analyze g(S) for g a smooth real-valued function on O(M), and then to specialise by setting g = f 0 71". 6). Choosing an orthonormal basis {Wi: i = 1, ... 9) the horizontal tangent vector field on the orthonormal frame bundle corresponding to the vector Wi.

The lower bound on curvature ensures that X does not explode. Com- parison arguments show that as t tends to 00 so lim inf r(Xt )! 2. Thus, X diverges to infinity. But the negative curvature means that the surface measure of the geodesic ball {x : r(x) = p} increases exponentially fast with p and the effect of this is that the diffusive component of e decreases rapidly. The drift of e can be controlle(j by techniques related to the existence of the lim sup bound and so e can be shown to 'freeze to a halt'.

The above theorem illustrates the basic strategy as propounded in Kendall (1981): one can deduce geometric implications in harmonic map theory by contrasting properties of BM(M) with properties of families of r -martingales in N. Equally, a property of Brownian motion or of r-martingales that leads to such implications is thereby interesting and worthy of further study. 2. LIMITING DIRECTIONS Theorem 9 can be mimicked for r -martingales of bounded dilatation. The next theorem follows by a lifting argument and by arguing as in Theorem 11 above.

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