By Mads Sielemann Jakobsen, Jakob Lemvig

http://www.sciencedirect.com/science/article/pii/S0022123615004152

We examine Gabor frames on in the neighborhood compact abelian teams with time–frequency shifts alongside non-separable, closed subgroups of the part house. Density theorems in Gabor research country helpful stipulations for a Gabor procedure to be a body or a Riesz foundation, formulated in basic terms when it comes to the index subgroup. within the classical effects the subgroup is believed to be discrete. We turn out density theorems for normal closed subgroups of the section area, the place the mandatory stipulations are given by way of the “size” of the subgroup. From those density effects we can expand the classical Wexler–Raz biorthogonal relatives and the duality precept in Gabor research to Gabor platforms with time–frequency shifts alongside non-separable, closed subgroups of the section area. Even within the euclidean surroundings, our effects are new.

Show description

Quick preview of Density and duality theorems for regular Gabor frames PDF

Similar Mathematics books

Symmetry: A Journey into the Patterns of Nature

Symmetry is throughout us. Our eyes and minds are attracted to symmetrical gadgets, from the pyramid to the pentagon. Of primary value to the way in which we interpret the area, this special, pervasive phenomenon shows a dynamic dating among items. In chemistry and physics, the concept that of symmetry explains the constitution of crystals or the speculation of basic debris; in evolutionary biology, the wildlife exploits symmetry within the struggle for survival; and symmetry—and the breaking of it—is imperative to rules in artwork, structure, and tune.

Combining a wealthy historic narrative along with his personal own trip as a mathematician, Marcus du Sautoy takes a distinct look at the mathematical brain as he explores deep conjectures approximately symmetry and brings us face-to-face with the oddball mathematicians, either previous and current, who've battled to appreciate symmetry's elusive traits. He explores what's possibly the main intriguing discovery to date—the summit of mathematicians' mastery within the field—the Monster, a big snowflake that exists in 196,883-dimensional area with extra symmetries than there are atoms within the solar.

what's it wish to clear up an old mathematical challenge in a flash of concept? what's it wish to be proven, ten mins later, that you've made a mistake? what's it prefer to see the realm in mathematical phrases, and what can that let us know approximately lifestyles itself? In Symmetry, Marcus du Sautoy investigates those questions and indicates mathematical rookies what it appears like to grapple with probably the most complicated rules the human brain can understand.

Do the Math: Secrets, Lies, and Algebra

Tess loves math simply because it is the one topic she will trust—there's consistently only one correct solution, and it by no means alterations. yet then she begins algebra and is brought to these pesky and mysterious variables, which appear to be far and wide in 8th grade. whilst even your folks and oldsters might be variables, how on the planet do you discover out the best solutions to the particularly vital questions, like what to do a couple of boy you love or whom to inform while a persons' performed whatever relatively undesirable?

Advanced Engineering Mathematics (2nd Edition)

This transparent, pedagogically wealthy ebook develops a powerful figuring out of the mathematical rules and practices that cutting-edge engineers want to know. both as powerful as both a textbook or reference guide, it techniques mathematical thoughts from an engineering point of view, making actual functions extra brilliant and mammoth.

Category Theory for the Sciences (MIT Press)

Type idea was once invented within the Forties to unify and synthesize diversified components in arithmetic, and it has confirmed remarkably profitable in permitting strong verbal exchange among disparate fields and subfields inside arithmetic. This booklet exhibits that class thought should be worthy outdoor of arithmetic as a rigorous, versatile, and coherent modeling language through the sciences.

Additional info for Density and duality theorems for regular Gabor frames

Show sample text content

Gröchenig, features of Gabor research on in the community compact abelian teams, in: Gabor research and Algorithms, in: utilized and Numerical Harmonic research, Birkhäuser Boston, Boston, MA, 1998, pp. 211–231. [21] ok. Gröchenig, Foundations of Time–Frequency research, utilized and Numerical Harmonic research, Birkhäuser Boston, Inc. , Boston, MA, 2001. [22] D. Han, D. Larson, body duality houses for projective unitary representations, Bull. Lond. Math. Soc. forty (4) (2008) 685–695. [23] D. Han, Y. Wang, Lattice tiling and the Weyl–Heisenberg frames, Geom. Funct. Anal. eleven (4) (2001) 742–758. [24] C. Heil, background and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl. thirteen (2) (2007) 113–166. [25] E. Hewitt, ok. A. Ross, summary Harmonic research. Vol. I: constitution of Topological teams. Integration idea, team Representations, moment version, Die Grundlehren der Mathematischen Wissenschaften, vol. a hundred and fifteen, Springer-Verlag, Berlin–New York, 1963. [26] E. Hewitt, ok. A. Ross, summary Harmonic research. Vol. II: constitution and research for Compact teams. research on in the neighborhood Compact Abelian teams, Die Grundlehren der Mathematischen Wissenschaften, vol. 152, Springer-Verlag, manhattan, 1970. [27] M. S. Jakobsen, J. Lemvig, Reproducing formulation for generalized translation invariant platforms on in the community compact abelian teams, Trans. Amer. Math. Soc. (2015), arXiv:1405. 4948, in press. [28] M. S. Jakobsen, J. Lemvig, Co-compact Gabor platforms on in the neighborhood compact abelian teams, J. Fourier Anal. Appl. , http://dx. doi. org/10. 1007/s00041-015-9407-0, arXiv:1411:4948, in press. [29] A. J. E. M. Janssen, Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl. 1 (4) (1995) 403–436. [30] R. A. Johnson, Atomic and nonatomic measures, Proc. Amer. Math. Soc. 25 (1970) 650–655. [31] G. Kaiser, A pleasant consultant to Wavelets, Birkhäuser Boston, Inc. , Boston, MA, 1994. [32] J. Ramanathan, T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (2) (1995) 148–153. [33] M. A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. forty (2) (1988) 257–338. [34] A. Ron, Z. Shen, Weyl–Heisenberg frames and Riesz bases in L2 (Rd ), Duke Math. J. 89 (2) (1997) 237–282. [35] D. T. Stoeva, O. Christensen, On R-duals and the duality precept in Gabor research, J. Fourier Anal. Appl. 21 (2) (2015) 383–400. [36] Z. -Q. Xiang, New characterizations of Riesz-type frames and balance of exchange duals of constant frames, Adv. Math. Phys. (2013) 1–11. [37] A. C. Zaanen, Integration, North-Holland Publishing Co. , Interscience Publishers John Wiley & Sons, Inc. , Amsterdam, manhattan, 1967, thoroughly revised variation of an advent to the speculation of integration.

Download PDF sample

Rated 4.80 of 5 – based on 46 votes