This distinct textual content is an advent to harmonic research at the least difficult symmetric areas, particularly Euclidean house, the sector, and the Poincaré higher part plane. This publication is meant for starting graduate scholars in arithmetic or researchers in physics or engineering. Written with an off-the-cuff type, the e-book areas an emphasis on motivation, concrete examples, historical past, and, exceptionally, functions in arithmetic, records, physics, and engineering.
Many corrections and updates were integrated during this new version. Updates contain 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, eventually, the author's comparisons of continuing conception with the finite analogues.
Topics featured in the course of the textual content contain inversion formulation for Fourier transforms, significant restrict theorems, Poisson's summation formulation and functions in crystallography and quantity thought, functions 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 functions to microwave engineering, basic domain names in H for discrete teams Γ, tessellations of H from such discrete crew activities, automorphic kinds, and the Selberg hint formulation and its functions in spectral conception in addition to quantity theory.
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