The Experts below are selected from a list of 14487 Experts worldwide ranked by ideXlab platform
Nir Sochen - One of the best experts on this subject based on the ideXlab platform.
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a wavelet plancherel theory with application to multipliers and sparse approximations
arXiv: Information Theory, 2017Co-Authors: Ron Levie, Nir SochenAbstract:We introduce an Extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space -- the so called window-signal space. There is a Canonical Extension of the wavelet transform to an isometric isomorphism between the window-signal space and the coefficient space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation in the coefficient space can be pulled-back to an operation in the window-signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window-signal space. As one example application, we show how continuous wavelet multipliers (also called Calder\'{o}n-Toeplitz Operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.
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a wavelet plancherel theory with application to sparse continuous wavelet transform
arXiv: Information Theory, 2017Co-Authors: Ron Levie, Nir SochenAbstract:We introduce a framework for calculating sparse approximations to signals based on elements of continuous wavelet systems. The method is based on an Extension of the continuous wavelet theory. In the new theory, the signal space is embedded in larger "abstract" signal space, which we call the window-signal space. There is a Canonical Extension of the wavelet transform on the window-signal space, which is an isometric isomorphism from the window-signal space to a space of functions on phase space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation on phase space can be pulled-back to an operation in the window-signal space. Using this pull back property, it is possible to pull back a search for big wavelet coefficients to the window-signal space. We can thus avoid inefficient calculations on phase space, performing all calculations entirely in the window-signal space. We consider in this paper a matching pursuit algorithm based on this coefficient search approach. Our method has lower computational complexity than matching pursuit algorithms based on a naive coefficient search.
Yuri A Neretin - One of the best experts on this subject based on the ideXlab platform.
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zak transform weil representation and integral operators with theta kernels
International Mathematics Research Notices, 2004Co-Authors: Tatiana Foth, Yuri A NeretinAbstract:The Weil representation of a real symplectic group Sp(2n, ℝ) admits a Canonical Extension to a holomorphic representation of a certain complex semigroup consisting of Lagrangian linear relations (this semigroup includes the Olshanskii semigroup). We obtain the explicit realization of the Weil representation of this semigroup in the Cartier model, that is, in the space of smooth sections of a certain line bundle on the 2n-dimensional torus T2n. We show that operators of the representation are integral operators whose kernels are theta-functions on T4n. We also extend this construction to a functor from a certain category of Lagrangian linear relations between symplectic vector spaces of different dimensions to a category of integral operators acting on sections of line bundles on the tori.
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zak transform weil representation and integral operators with theta kernels
arXiv: Classical Analysis and ODEs, 2003Co-Authors: Tatiana Foth, Yuri A NeretinAbstract:The Weil representation of a real symplectic group $Sp(2n,R)$ admits a Canonical Extension to a holomorphic representation of a certain complex semigroup consisting of Lagrangian linear relations (this semigroup includes the Olshanski semigroup). We obtain the explicit realization of the Weil representation of this semigroup in the Cartier model, i.e., in the space of smooth sections of a certain line bundle on the $2n$-dimensional torus $T^{2n}$. We show that operators of the representation are integral operators whose kernels are theta-functions on $T^{4n}$.
Ron Levie - One of the best experts on this subject based on the ideXlab platform.
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a wavelet plancherel theory with application to multipliers and sparse approximations
arXiv: Information Theory, 2017Co-Authors: Ron Levie, Nir SochenAbstract:We introduce an Extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space -- the so called window-signal space. There is a Canonical Extension of the wavelet transform to an isometric isomorphism between the window-signal space and the coefficient space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation in the coefficient space can be pulled-back to an operation in the window-signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window-signal space. As one example application, we show how continuous wavelet multipliers (also called Calder\'{o}n-Toeplitz Operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.
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a wavelet plancherel theory with application to sparse continuous wavelet transform
arXiv: Information Theory, 2017Co-Authors: Ron Levie, Nir SochenAbstract:We introduce a framework for calculating sparse approximations to signals based on elements of continuous wavelet systems. The method is based on an Extension of the continuous wavelet theory. In the new theory, the signal space is embedded in larger "abstract" signal space, which we call the window-signal space. There is a Canonical Extension of the wavelet transform on the window-signal space, which is an isometric isomorphism from the window-signal space to a space of functions on phase space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation on phase space can be pulled-back to an operation in the window-signal space. Using this pull back property, it is possible to pull back a search for big wavelet coefficients to the window-signal space. We can thus avoid inefficient calculations on phase space, performing all calculations entirely in the window-signal space. We consider in this paper a matching pursuit algorithm based on this coefficient search approach. Our method has lower computational complexity than matching pursuit algorithms based on a naive coefficient search.
Mai Gehrke - One of the best experts on this subject based on the ideXlab platform.
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δ1 completions of a poset
Order, 2013Co-Authors: Mai Gehrke, Ramon Jansana, Alessandra PalmigianoAbstract:A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A Δ1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that Δ1-completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of up-sets of the poset, and a binary relation between these two systems. Certain Δ1-completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact Δ1-completions of a poset include its MacNeille completion and all its join- and all its meet-completions. These completions also include the Canonical Extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, we use our parametric description of Δ1-completions to compare the Canonical Extension to other compact Δ1-completions identifying its relative merits.
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bounded distributive lattice expansions
Mathematica Scandinavica, 2004Co-Authors: Mai Gehrke, Bjarni JonssonAbstract:A new notion of a Canonical Extension $\mathbf{A}^{\sigma }$ is introduced that applies to arbitrary bounded distributive lattice expansions (DLEs) $\mathbf{A} $. The new definition agrees with the earlier ones whenever they apply. In particular, for a bounded distributive lattice $\mathbf{A}, \mathbf{A}^{\sigma }$ has the same meaning as before. A novel feature is the introduction of several topologies on the universe of the Canonical Extension of a DL. One of these topologies is used to define the Canonical Extension $f^{\sigma }:\mathbf{A}^{\sigma }\rightarrow \mathbf{B}^{\sigma }$ of an arbitrary map $f:\mathbf{A}\rightarrow \mathbf{B}$ between DLs, and hence to define the Canonical Extension $\mathbf{A}^{\sigma }$ of an arbitrary DLE $\mathbf{A}$. Together the topologies form a powerful tool for showing that many properties of DLEs are preserved by Canonical Extensions.
Tatiana Foth - One of the best experts on this subject based on the ideXlab platform.
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zak transform weil representation and integral operators with theta kernels
International Mathematics Research Notices, 2004Co-Authors: Tatiana Foth, Yuri A NeretinAbstract:The Weil representation of a real symplectic group Sp(2n, ℝ) admits a Canonical Extension to a holomorphic representation of a certain complex semigroup consisting of Lagrangian linear relations (this semigroup includes the Olshanskii semigroup). We obtain the explicit realization of the Weil representation of this semigroup in the Cartier model, that is, in the space of smooth sections of a certain line bundle on the 2n-dimensional torus T2n. We show that operators of the representation are integral operators whose kernels are theta-functions on T4n. We also extend this construction to a functor from a certain category of Lagrangian linear relations between symplectic vector spaces of different dimensions to a category of integral operators acting on sections of line bundles on the tori.
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zak transform weil representation and integral operators with theta kernels
arXiv: Classical Analysis and ODEs, 2003Co-Authors: Tatiana Foth, Yuri A NeretinAbstract:The Weil representation of a real symplectic group $Sp(2n,R)$ admits a Canonical Extension to a holomorphic representation of a certain complex semigroup consisting of Lagrangian linear relations (this semigroup includes the Olshanski semigroup). We obtain the explicit realization of the Weil representation of this semigroup in the Cartier model, i.e., in the space of smooth sections of a certain line bundle on the $2n$-dimensional torus $T^{2n}$. We show that operators of the representation are integral operators whose kernels are theta-functions on $T^{4n}$.