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The Theory of Constructive Signal Analysis
Author(s) -
Benedetto John J.
Publication year - 1981
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm198165137
Subject(s) - semigroup , mathematics , constructive , closure (psychology) , constructive proof , variety (cybernetics) , pure mathematics , invariant (physics) , discrete mathematics , algebra over a field , mathematical analysis , mathematical physics , computer science , statistics , process (computing) , economics , market economy , operating system
Let h ( x ) = e −αx k ( x ), where and λ0=0. The closure theorem, V h = L 1 (ℝ), is proved for various α and k ( V h is the L 1 ‐closed variety generated by h ). The Tauberian condition, | ĥ | > 0, is not used, since generally this condition is difficult to compute directly. The functions h arise naturally in time series and analytic number theory. The technique of proof is constructive and depends on the semigroup {γ j } generated by {λ j }. The semigroup theory which consolidates and completes the results herein will be developed separately as “A closure problem for signals in semigroup invariant systems.”