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

Li, Yun‐Zhang; Zhang, Wei

doi:10.1007/s11425-018-9468-8

Cited by 11 publications

(13 citation statements)

References 47 publications

(47 reference statements)

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“…Proof By a standard argument, we have

\begin{matrix} true \sum_{l = 1}^{L} {(||, {normalΘ}_{a} χ_{S_{l}} false(x, ξ false))}^{2} = & {(||, \sum_{j \in ℤ} a^{\frac{j}{2}} χ_{S_{1}} false(a^{j} x false) e^{- 2 π i j ξ})}^{2} \\ = & false| F false(ξ false) {false|}^{2} \end{matrix}

for

false(x, ξ false) \in E \times false[0, 1 false)

with

F (ξ) = \sum_{j = - \infty}^{n} a^{\frac{j}{2}} e^{- 2 π i j ξ} - \sum_{j = 1}^{k} a^{\frac{n_{j}}{2}} e^{- 2 π i n_{j} ξ},

which satisfies

F (\frac{1}{2}) = 0 .

Similarly to Theorem 4.3, we can prove that

scriptM scriptD false(normalΨ, a false)

is not a frame for

L^{2} false(ℝ_{+} false)

by Li and Zhang …”

Section: Frame Setsmentioning

confidence: 99%

“…So we only need to prove that there exists

A > 0

such that

\begin{matrix} true \sum_{l = 1}^{L} {(||, {normalΘ}_{a} χ_{S_{l}} false(\cdot, \cdot false))}^{2} \geq A \end{matrix}

a.e. on

false[1, a false) \times false[0, 1 false)

by Li and Zhang 11, Theorem 2.7 . Observe that

false{E_{k} : k \in I false}

is a finite partition of

false[1, a false)

, we only need to prove that, for every

k \in I

, there exists a constant

A false(k false) > 0

such that

| Θ_{a} χ_{S_{l_{k}}} (\cdot, \cdot) | \geq A (k) a.e.

…”

Section: Frame Setsmentioning

confidence: 99%

“…All these make studying

scriptM scriptD

‐frames nontrivial and challenging to some extent. In previous works, 8,10–12 the

{normalΘ}_{a}

‐transform was introduced and applied to studying

scriptM scriptD

‐frames for

L^{2} false(ℝ_{+} false)

. Li and Zhang 11 characterized and expressed

scriptM scriptD

‐frames and

scriptM scriptD

‐dual frames of the form ().…”

Section: Introductionmentioning

confidence: 99%

“…

false(x, ξ false) \in ℝ_{+} \times ℝ

. By Li and Zhang, 11, Lemma 2.3

{normalΘ}_{a}

‐transform is a unitary operator from

L^{2} false(ℝ_{+} false)

onto

L^{2} false(false[1, a false) \times false[0, 1 false) false)

. From Li and Zhang 11, Theorem 3.9 and Wang and Li, 12, Corollary 3.1 we know that a frame

scriptM scriptD false(normalΨ, a false)

for

L^{2} false(ℝ_{+} false)

is a Riesz basis for

L^{2} false(ℝ_{+} false)

if and only if the cardinality

normalcard false(normalΨ false)

normalΨ

equals 1.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dilation‐and‐modulation frame sets on the half real line

2022

Math Methods in App Sciences

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Wavelet and Gabor frame sets on the real line have been extensively studied. Due to it not being a group under addition, the half real line admits no traditional wavelet and Gabor frames as the real line. In this paper, we introduce a class of dilation‐and‐modulation frame sets ( scriptMscriptD‐frame sets) on the half real line. We characterize scriptMscriptD‐frame sets and scriptMscriptD‐tight frame sets and obtain some sufficient conditions for scriptMscriptD‐tight frame sets. In particular, we prove that scriptMscriptD‐tight frame sets must be bounded, but scriptMscriptD‐frame sets need not be bounded. Some examples are also provided.

show abstract

“…Proof By a standard argument, we have

\begin{matrix} true \sum_{l = 1}^{L} {(||, {normalΘ}_{a} χ_{S_{l}} false(x, ξ false))}^{2} = & {(||, \sum_{j \in ℤ} a^{\frac{j}{2}} χ_{S_{1}} false(a^{j} x false) e^{- 2 π i j ξ})}^{2} \\ = & false| F false(ξ false) {false|}^{2} \end{matrix}

for

false(x, ξ false) \in E \times false[0, 1 false)

with

F (ξ) = \sum_{j = - \infty}^{n} a^{\frac{j}{2}} e^{- 2 π i j ξ} - \sum_{j = 1}^{k} a^{\frac{n_{j}}{2}} e^{- 2 π i n_{j} ξ},

which satisfies

F (\frac{1}{2}) = 0 .

Similarly to Theorem 4.3, we can prove that

scriptM scriptD false(normalΨ, a false)

is not a frame for

L^{2} false(ℝ_{+} false)

by Li and Zhang …”

Section: Frame Setsmentioning

confidence: 99%

“…So we only need to prove that there exists

A > 0

such that

\begin{matrix} true \sum_{l = 1}^{L} {(||, {normalΘ}_{a} χ_{S_{l}} false(\cdot, \cdot false))}^{2} \geq A \end{matrix}

a.e. on

false[1, a false) \times false[0, 1 false)

by Li and Zhang 11, Theorem 2.7 . Observe that

false{E_{k} : k \in I false}

is a finite partition of

false[1, a false)

, we only need to prove that, for every

k \in I

, there exists a constant

A false(k false) > 0

such that

| Θ_{a} χ_{S_{l_{k}}} (\cdot, \cdot) | \geq A (k) a.e.

…”

Section: Frame Setsmentioning

confidence: 99%

“…All these make studying

scriptM scriptD

‐frames nontrivial and challenging to some extent. In previous works, 8,10–12 the

{normalΘ}_{a}

‐transform was introduced and applied to studying

scriptM scriptD

‐frames for

L^{2} false(ℝ_{+} false)

. Li and Zhang 11 characterized and expressed

scriptM scriptD

‐frames and

scriptM scriptD

‐dual frames of the form ().…”

Section: Introductionmentioning

confidence: 99%

“…

false(x, ξ false) \in ℝ_{+} \times ℝ

. By Li and Zhang, 11, Lemma 2.3

{normalΘ}_{a}

‐transform is a unitary operator from

L^{2} false(ℝ_{+} false)

onto

L^{2} false(false[1, a false) \times false[0, 1 false) false)

. From Li and Zhang 11, Theorem 3.9 and Wang and Li, 12, Corollary 3.1 we know that a frame

scriptM scriptD false(normalΨ, a false)

for

L^{2} false(ℝ_{+} false)

is a Riesz basis for

L^{2} false(ℝ_{+} false)

if and only if the cardinality

normalcard false(normalΨ false)

normalΨ

equals 1.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dilation‐and‐modulation frame sets on the half real line

2022

Math Methods in App Sciences

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show abstract

“…Li and Zhang in [22] characterized F a -frames, F a -dual frames and Parseval F a -frames for L 2 (R + ) of the form {a k 2 ψ(a k •)} k∈Z , and as a special case, Li and Wang studied F a -frame sets in [21]. Its multi-window and vector-valued cases and another variation were studied in [20,23,24,27]. By [22,Corollary 3.1], for 0 = ψ ∈ L 2 (R + ), the following are equivalent:…”

Section: Definition 11 Givenmentioning

confidence: 99%

The equivalence of $F_{a}$-frames

Hussain

2020

J Inequal Appl

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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.

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A Survey on the Time-Frequency Analysis on the Half Real Line

2021

Trends in Mathematics

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

Cited by 11 publications

References 47 publications

Dilation‐and‐modulation frame sets on the half real line

Dilation‐and‐modulation frame sets on the half real line

The equivalence of $F_{a}$-frames

A Survey on the Time-Frequency Analysis on the Half Real Line

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