By Audrey Terras
This exact textual content is an advent to harmonic research at the least difficult symmetric areas, specifically Euclidean house, the sector, and the Poincaré higher part plane. This e-book is meant for starting graduate scholars in arithmetic or researchers in physics or engineering. Written with an off-the-cuff sort, the publication locations an emphasis on motivation, concrete examples, heritage, and, notably, purposes in arithmetic, facts, physics, and engineering.
Many corrections and updates were included during this re-creation. 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 listen the form of a drum?", A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, and, eventually, the author's comparisons of constant concept with the finite analogues.
Topics featured during the textual content contain inversion formulation for Fourier transforms, critical restrict theorems, Poisson's summation formulation and purposes in crystallography and quantity idea, functions of round harmonic research to the hydrogen atom, the Radon rework, 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 staff activities, automorphic varieties, and the Selberg hint formulation and its purposes in spectral conception in addition to quantity theory.