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Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. TheParseval framesare particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles—this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors—and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of -bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.
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On Parseval Frames of Kernel Functions in de Branges Spaces of Entire Vector Valued Functions
We consider the existence and structure properties of Parseval frames of kernel functions in vector valued de Branges spaces. We develop some sufficient conditions for Parseval sequences by identifying the main construction with Naimark dilation of frames. The dilation occurs by embedding the de Branges space of vector valued functions into a dilated de Branges space of vector valued functions. The embedding also maps the kernel functions associated with a frame sequence of the original space into a Riesz basis for the embedding space. We also develop some sufficient conditions for a dilated de Branges space to have the Kramer sampling property.
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- PAR ID:
- 10314053
- Date Published:
- Journal Name:
- New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative
- Volume:
- 286
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-76473-9_1
