Sparsity and spectral properties of dual frames

Krahmer, Felix; Kutyniok, Gitta und Lemvig, Jakob (15. August 2013): Sparsity and spectral properties of dual frames. In: Linear Algebra and its Applications, Vol. 439, No. 4: pp. 982-998

Abstract

We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than non-vanishing entries, where n denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a generalized notion of spark. We then study the spectral properties of dual frames in terms of singular values of the synthesis operator. We provide a complete characterization for which spectral patterns of dual frames are possible for a fixed frame. For many cases, we provide simple explicit constructions for dual frames with a given spectrum, in particular, if the constraint on the dual is that it be tight.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Bavarian Chair for Mathematical Foundations of Artificial Intelligence
• Subjects: 500 Science > 510 Mathematics
• ISSN: 00243795
• Language: English