The power of group theoretic ideology is successfully illustrated by this wide range of topics it is widely understood now that the explanation of this miraculous power stems from group theoretic ideas underlying practically everything in harmonic analysis. Classical harmonic analysis is an important part of modern physics and mathematics comparable in its significance with calculus created in the 18th and 19th centuries as a distinct mathematical discipline it continued to develop and still does conquering new unexpected areas and producing . In mathematics noncommutative harmonic analysis is the field in which results from fourier analysis are extended to topological groups that are not commutative since locally compact abelian groups have a well understood theory pontryagin duality which includes the basic structures of fourier series and fourier transforms the major business of non commutative harmonic analysis is usually . Ground for group theory harmonic analysis and differential geometry and it even has some points of contact with number theory and mathematical physics it is fascinating to see the interplay between these areas as illustrated by an abundance of interesting examples there are two distinct approaches to the theory of commutative spaces ana. Note citations are based on reference standards however formatting rules can vary widely between applications and fields of interest or study the specific requirements or preferences of your reviewing publisher classroom teacher institution or organization should be applied
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