Further generalisation to locally compact abelian groups is required in number theory.In non-commutative harmonic analysis, the idea is taken even further in the Selberg trace formula, but takes on a much deeper character.…

Stone's Theorem can be recast using the language of the Fourier transform.The real line is a locally compact abelian group.Non-degenerate *-representations of the group C*-algebra are in one-to-one correspondence with strongly continuous unitary representations of, i.e., strongly continuous one-parameter unitary groups.…

A locally compact abelian group G is compact if and only if the dual group G^ is discrete.Conversely, G is discrete if and only if G^ is compact.…

However, if the group G is a separable locally compact abelian group, then the dual group is metrizable.

More precisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups.The study of locally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelian locally compact groups.…

There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly.…

The resulting theory is a central part of harmonic analysis.The representation theory for locally compact abelian groups is described by Pontryagin duality.By homogeneity, local compactness for a topological group need only be checked at the identity.…

This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin inversion theorem).It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity.…

It is useful to regard the dual group functorially.In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms.…

If G is a Lie group but not compact nor abelian, this is a difficult matter of harmonic analysis.The locally compact abelian case is part of the Pontryagin duality theory.…

Thus any non-locally compact example of Pontryagin duality is a group where the natural evaluation pairing of G and G^ is not continuous.The following books have chapters on locally compact abelian groups, duality and Fourier transform.The Dixmier reference (also available in English translation) has material on non-commutative harmonic analysis.…

Depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem.See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform.…

The character group of the cyclic group also appears in the theory of the discrete Fourier transform.For locally compact abelian groups, the character group (with an assumption of continuity) is central to Fourier analysis.…

In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters.Equivalently, a suitably defined dual set is relatively dense in the Pontryagin dual of the group.…

The Hausdorff-Young inequality can also be established for the Fourier transform on locally compact abelian groups.We also note that the norm estimate of 1 is not optimal.…

However, the L p -norm on this space depends on the choice of Haar measure, so if one wants to talk about isometries it is important to keep track of the Haar measure being used.The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform.…

One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context.Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).…

He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan.Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra.…

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact.More precisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups.The study of locally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelian locally compact groups.…

The self-duality of the adeles of the function field of a curve over a finite field easily implies the Riemann-Roch theorem for the curve and the duality theory for the curve.As a locally compact abelian group, the adeles have a nontrivial translation invariant measure.Similarly, the group of ideles has a nontrivial translation invariant measure using which one defines a zeta integral.…