SHIFT INVARIANT SPACES FOR LOCAL FIELDS

2011 ◽  
Vol 09 (03) ◽  
pp. 417-426 ◽  
Author(s):  
A. AHMADI ◽  
A. ASKARI HEMMAT ◽  
R. RAISI TOUSI
Keyword(s):  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Open Subgroup ◽  
Local Fields ◽  
Locally Compact Abelian Group ◽  
Compact Open Subgroup ◽  
Parseval Frame ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

This paper is an investigation of shift invariant subspaces of L2(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L2(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L2(G) is a direct sum of principle shift invariant subspaces of L2(G) generated by Parseval frame generators.

MULTI-CHANNEL SAMPLING ON SHIFT-INVARIANT SPACES WITH FRAME GENERATORS

2012 ◽  
Vol 10 (01) ◽  
pp. 1250003 ◽  
Author(s):  
A. G. GARCIA ◽  
J. M. KIM ◽  
K. H. KWON ◽  
G. J. YOON
Keyword(s):  
Continuous Function ◽  
Sufficient Conditions ◽  
Sampling Theory ◽  
Invariant Space ◽  
Frame Sequence ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces ◽  
Sampling Expansions ◽  
Shift Invariant Space

Let φ be a continuous function in L2(ℝ) such that the sequence {φ(t - n)}n∈ℤ is a frame sequence in L2(ℝ) and assume that the shift-invariant space V(φ) generated by φ has a multi-banded spectrum σ(V). The main aim in this paper is to derive a multi-channel sampling theory for the shift-invariant space V(φ). By using a type of Fourier duality between the spaces V(φ) and L2[0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V(φ).

SAMPLING EXPANSION IN SHIFT INVARIANT SPACES

2008 ◽  
Vol 06 (02) ◽  
pp. 223-248 ◽  
Author(s):  
J. M. KIM ◽  
K. H. KWON
Keyword(s):  
Truncation Error ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Expansion Formula ◽  
Sampling Series ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

For any ϕ(t) in L2(ℝ), let V(ϕ) be the closed shift invariant subspace of L2(ℝ) spanned by integer translates {ϕ(t - n) : n ∈ ℤ} of ϕ(t). Assuming that ϕ(t) is a frame or a Riesz generator of V(ϕ), we first find conditions under which V(ϕ) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(ϕ) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L2(ℝ) that belongs to a sampling subspace of L2(ℝ). Several illustrating examples are also provided.

Frames associated with shift-invariant spaces on local fields

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3097-3110 ◽  
Author(s):  
Firdous Shah ◽  
Owais Ahmad ◽  
Asghar Rahimi
Keyword(s):  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Necessary Condition ◽  
Wavelet Frames ◽  
Unified Approach ◽  
Shift Invariant Spaces ◽  
Invariant Spaces

In this paper, we present a unified approach to the study of shift-invariant systems to be frames on local fields of positive characteristic. We establish a necessary condition and three sufficient conditions under which the shift-invariant systems on local fields constitute frames for L2(K). As an application of these results, we obtain some known conclusions about the Gabor frames and wavelet frames on local fields.

CHANNELED SAMPLING IN SHIFT INVARIANT SPACES

2007 ◽  
Vol 05 (05) ◽  
pp. 753-767 ◽  
Author(s):  
Y. M. HONG ◽  
J. M. KIM ◽  
K. H. KWON ◽  
E. H. LEE
Keyword(s):  
Closed Subspace ◽  
Single Channel ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

We develop sampling expansion formulas on the shift invariant closed subspace V(ϕ) of L2ℝ generated by a frame or a Riesz generator ϕ(t). We find necessary and sufficient conditions under which a regular shifted sampling expansion to hold on V(ϕ) and also introduce a single channel sampling on V(ϕ) together with some illustrating examples.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

Characterization of Low-pass Filters on Local Fields of Positive Characteristic

2016 ◽  
Vol 59 (3) ◽  
pp. 528-541 ◽  
Author(s):  
Qaiser Jahan
Keyword(s):  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Necessary And Sufficient ◽  
Low Pass Filters ◽  
The Scaling Function

AbstractIn this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field K of positive characteristic associated with the scaling function for multiresolution analysis of L2(K). We use probability and martingale methods to provide such a characterization.

scholarly journals Multigenerator Gabor Frames on Local Fields

2018 ◽  
Vol 33 (2) ◽  
pp. 307
Author(s):  
Owais Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Gabor System ◽  
Necessary And Sufficient

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

The Integral Extension of Isometries of Quadratic Forms Over Local Fields

1966 ◽  
Vol 18 ◽  
pp. 920-942 ◽  
Author(s):  
Allan Trojan
Keyword(s):  
Vector Space ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
The Other ◽  
Quadratic Space ◽  
Integral Extension ◽  
Ring Of Integers ◽  
Extension Of Isometries ◽  
Necessary And Sufficient

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

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