compact abelian group造句

• The theorem also holds more generally in locally compact abelian groups.
• The representation theory for locally compact abelian groups is described by Pontryagin duality.
• The real line \ mathbf { R } is a locally compact abelian group.
• The following books have chapters on locally compact abelian groups, duality and Fourier transform.
• It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.
• In the special case when " R " is the ring of locally compact abelian group.
• For a locally compact abelian group " G ", every irreducible unitary representation has dimension 1.
• A similar result is true when "'R "'is replaced by any locally compact abelian group.
• *Suppose that G is a locally compact abelian group that contains the unit circle \ mathbb T as a subgroup.
• This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is "'R " '.
• The "'Pontryagin duality theorem "'itself states that locally compact abelian groups identify naturally with their bidual.
• This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and Andr?Weil in 1940.
• In this case, the unitary dual \ hat { G } is a group, in fact another locally compact abelian group.
• The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform.
• In what follows, "'LCA "'is the category of locally compact abelian groups and continuous group homomorphisms.
• If G is a locally compact " abelian " group, a separable locally compact abelian group, then the dual group is metrizable.
• The Mellin transform may also be viewed as the Gelfand transform for the convolution algebra of the locally compact abelian group of positive real numbers with multiplication.
• On locally compact abelian groups, a version of the convolution theorem holds : the Fourier transform of a convolution is the pointwise product of the Fourier transforms.
• There is a canonical isomorphism G \ cong \ widehat { \ widehat { G } } between any locally compact abelian group G and its double dual.
• The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934.