Convolution, Banach Algebras, and the Uncertainty Principle

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


There is a bilinear map on V G (i.e., a map from V G × V G to V G which is linear in each variable) which becomes the pointwise product of functions under the FT. This map is called the convolution of functions for the FT. The inner product space V G together with the convolution form a Banach algebra. The discussion of convolution and Banach algebras comprises the first two sections of this chapter. In the final section, we prove that the order of G does not exceed the product of the cardinalities of the supports of f and \(\hat{f}\) provided f is nonzero. This result is known as the uncertainty principle.


Abelian Group Banach Algebra Uncertainty Principle Linear Functional Character Basis 
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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of DefenseWashington

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