A new function-valued inner product and corresponding function-valued frames in<i>L</i><sub>2</sub>(0,∞)

Fard, M. A. Hasankhani; Dehghan, Mohammad Ali

doi:10.1080/03081087.2013.801968

Cited by 7 publications

(6 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The system (1.6) is related to a kind of function-valued frames in [25] by Hasankhani and Dehghan. They introduced the notion of function-valued frame as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Take f j = D a j ψ for some ψ ∈ L 2 (R + ). Then, applying [25,Theorem 4.8], we have that {D a j ψ} j∈Z is a function-valued frame for L 2 (R + ) with respect to a if and only if MD(ψ, a) (i.e., L = 1 in (1.6)) is a frame for L 2 (R + ). We characterized frames of the form MD(ψ, a) in [37] in terms of the bi-infinite matrix-valued function G(ψ, •) = (D a j+l ψ(•)) j, l∈Z , where the notion of Θ a -transform was not formally formulated.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multi-window dilation-and-modulation frames on the half real line

Zhang

2020

Sci. China Math.

View full text Add to dashboard Cite

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively, and have been studied extensively. However, dilation-and-modulation systems cannot be derived from wavelet or Gabor systems. This study aims to investigate a class of dilation-and-modulation systems in the causal signal space L 2 (R + ). L 2 (R + ) can be identified as a subspace of L 2 (R), which consists of all L 2 (R)-functions supported on R + but not closed under the Fourier transform. Therefore, the Fourier transform method does not work in L 2 (R + ). Herein, we introduce the notion of Θa-transform in L 2 (R + ) and characterize the dilation-and-modulation frames and dual frames in L 2 (R + ) using the Θa-transform; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L 2 (R + ). Furthermore, it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than 1.Finally, some examples are provided to illustrate the generality of our results.

show abstract

“…The system (1.6) is related to a kind of function-valued frames in [25] by Hasankhani and Dehghan. They introduced the notion of function-valued frame as follows.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multi-window dilation-and-modulation frames on the half real line

Zhang

2020

Sci. China Math.

View full text Add to dashboard Cite

show abstract

“…Motivated by this observation, some mathematicians studied Walsh series-based wavelet analysis in L 2 (R + ) using Cantor group operation on R + ( [1, 6-9, 16, 17]). Recently, Hasankhani Fard and Dehghan in [12] introduced the notion of function-valued frame in L 2 (R + ) which is referred to as "F a -frame" in our papers. Let us first recall and extend some related notions.…”

Section: Introductionmentioning

confidence: 99%

“…The following definition is an extension of [12,Definition 2.1], and that in [23] which only dealt with functions in L 2 (R + ). It is slightly different from [12, Definition 2.1], even for functions in L 2 (R + ), but it is more convenient for our purpose.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The equivalence of $F_{a}$-frames

Hussain

2020

J Inequal Appl

View full text Add to dashboard Cite

Structured frames such as wavelet and Gabor frames in L 2 (R) have been extensively studied. But L 2 (R +) cannot admit wavelet and Gabor systems due to R + being not a group under addition. In practice, L 2 (R +) models the causal signal space. The function-valued inner product-based F a-frame for L 2 (R +) was first introduced by Hasankhani Fard and Dehghan, where an F a-frame was called a function-valued frame. In this paper, we introduce the notions of F a-equivalence and unitary F a-equivalence between F a-frames, and present a characterization of the F a-equivalence and unitary F a-equivalence. This characterization looks like that of equivalence and unitary equivalence between frames, but the proof is nontrivial due to the particularity of F a-frames.

show abstract