By Audrey Terras

This distinctive textual content is an advent to harmonic research at the easiest symmetric areas, specifically Euclidean house, the field, and the Poincaré top part plane. This booklet is meant for starting graduate scholars in arithmetic or researchers in physics or engineering. Written with a casual variety, the publication areas an emphasis on motivation, concrete examples, historical past, and, specifically, purposes in arithmetic, data, physics, and engineering.

Many corrections and updates were included during this new version. Updates comprise discussions of P. Sarnak and others' paintings on quantum chaos, the paintings of T. Sunada, Marie-France Vignéras, Carolyn Gordon, and others on Mark Kac's query "Can you pay attention the form of a drum?", A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, and, ultimately, the author's comparisons of constant concept with the finite analogues.

Topics featured in the course of the textual content contain inversion formulation for Fourier transforms, crucial restrict theorems, Poisson's summation formulation and purposes in crystallography and quantity conception, purposes of round harmonic research to the hydrogen atom, the Radon remodel, non-Euclidean geometry at the Poincaré top part airplane H or unit disc and purposes to microwave engineering, basic domain names in H for discrete teams Γ, tessellations of H from such discrete crew activities, automorphic varieties, and the Selberg hint formulation and its functions in spectral conception in addition to quantity theory.

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