Frame duality properties for projective unitary representations |
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Authors: | Han Deguang; Larson David |
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Institution: | Department of Mathematics University of Central Florida Orlando, FL 32816 USA |
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Abstract: | Let be a projective unitary representation of a countable groupG on a separable Hilbert space H. If the set B of Bessel vectorsfor is dense in H, then for any vector x H the analysis operatorx makes sense as a densely defined operator from B to 2(G)-space.Two vectors x and y are called -orthogonal if the range spacesof x and y are orthogonal, and they are -weakly equivalent ifthe closures of the ranges of x and y are the same. These propertiesare characterized in terms of the commutant of the representation.It is proved that a natural geometric invariant (the orthogonalityindex) of the representation agrees with the cyclic multiplicityof the commutant of (G). These results are then applied to Gaborsystems. A sample result is an alternate proof of the knowntheorem that a Gabor sequence is complete in L2( d) ifand only if the corresponding adjoint Gabor sequence is 2-linearlyindependent. Some other applications are also discussed. |
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