Physics Experiments and Abstract Harmonic Analysis: A Generalized Peano Kernel Theorem for Estimating Errors in Laser Interferometry

Document Type

Presentation Abstract

Presentation Date

10-8-2012

Abstract

Photonic Doppler Velocimetry (PDV) is a measurement of the Doppler shift of laser light reflecting off of a moving surface, and it is usually used to determine moving surface positions under extreme temperatures, pressures, and energies. The actual measurements are voltages on a digital oscilloscope, which must be mathematically processed to back out the changing position of the surface. Position is usually parameterized by local polynomial fitting, and it is important to have some estimates of the "error bars" of the fit. Seemingly unrelated, the Peano Kernel Theorem is a century-old result giving a special kernel formula for numerical quadrature schemes, but it turns out that it can be adapted to give error estimates for local polynomial fits, which makes it relevant for PDV analysis. The proof of the generalized result uses basic Fourier analysis, but finding the right setting in which the "easy" proof actually holds requires the theory of Laplace transformable tempered distributions. We will describe the experimental application driving this work, the proof, and the abstract harmonic analysis needed to formulate the generalized Peano Kernel Theorem.

Additional Details

Monday, 8 October 2012
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

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