The aim of this work is to study (Multi-window) Gabor systems in the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell ^2(\mathbb {Z} \times \mathbb {Z}, \mathbb {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo>×</mo> <mi mathvariant="double-struck">Z</mi> <mo>,</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, denoted by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {G}(g,L,M,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and defined by: <Equation ID="Equ6"> <EquationSource Format="TEX">$\left\{ (k_1,k_2)\in \mathbb {Z}^2\mapsto e^{2\pi i \frac{m_1}{M}k_1} g_l(k - nN) e^{2\pi j \frac{m_2}{M}k_2} \right\} _{l \in \mathbb {N}_L, (m_1, m_2) \in \mathbb {N}_M^2, n \in \mathbb {Z}^2},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mfenced close="}" open="{"> <mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> <mo>↦</mo> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mfrac> <msub> <mi>m</mi> <mn>1</mn> </msub> <mi>M</mi> </mfrac> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> </msup> <msub> <mi>g</mi> <mi>l</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>-</mo> <mi>n</mi> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>j</mi> <mfrac> <msub> <mi>m</mi> <mn>2</mn> </msub> <mi>M</mi> </mfrac> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </msup> </mfenced> <mrow> <mi>l</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mi>L</mi> </msub> <mo>,</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">N</mi> <mi>M</mi> <mn>2</mn> </msubsup> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </mrow> </msub> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where, L, M, N are positive integers, i, j are the imaginary units in the quaternion algebra, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \{g_l\}_{l \in \mathbb {N}_L} \subset \ell ^2(\mathbb {Z} \times \mathbb {Z}, \mathbb {H}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>g</mi> <mi>l</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>l</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mi>L</mi> </msub> </mrow> </msub> <mo>⊂</mo> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo>×</mo> <mi mathvariant="double-struck">Z</mi> <mo>,</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Special emphasis is placed on the case where the sequences <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g_l$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>l</mi> </msub> </math></EquationSource> </InlineEquation> are real-valued. The questions addressed in this work include the characterization of quaternionic Gabor systems that form frames, the characterization of those that are orthonormal bases, and the admissibility of such systems. We also explore necessary and/or sufficient conditions for Gabor frames. The issue of duality is also discussed. Furthermore, we study the stability of these systems.

Discrete Quaternionic (Multi-window) Gabor Systems

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发表日期：

2025年11月6日

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Najib Khachiaa

摘要

The aim of this work is to study (Multi-window) Gabor systems in the space $\ell ^2(\mathbb {Z} \times \mathbb {Z}, \mathbb {H})$ $ℓ^{2} (Z \times Z, H)$ , denoted by $\mathcal {G}(g,L,M,N)$ $G (g, L, M, N)$ , and defined by: $\left\{ (k_1,k_2)\in \mathbb {Z}^2\mapsto e^{2\pi i \frac{m_1}{M}k_1} g_l(k - nN) e^{2\pi j \frac{m_2}{M}k_2} \right\} _{l \in \mathbb {N}_L, (m_1, m_2) \in \mathbb {N}_M^2, n \in \mathbb {Z}^2},$ ${\{(k_{1}, k_{2}) \in Z^{2} \mapsto e^{2 π i \frac{m_{1}}{M} k_{1}} g_{l} (k - n N) e^{2 π j \frac{m_{2}}{M} k_{2}}\}}_{l \in N_{L}, (m_{1}, m_{2}) \in N_{M}^{2}, n \in Z^{2}},$ where, L, M, N are positive integers, i, j are the imaginary units in the quaternion algebra, and $ \{g_l\}_{l \in \mathbb {N}_L} \subset \ell ^2(\mathbb {Z} \times \mathbb {Z}, \mathbb {H}) $ ${g_{l}}_{l \in N_{L}} \subset ℓ^{2} (Z \times Z, H)$ . Special emphasis is placed on the case where the sequences $g_l$ $g_{l}$ are real-valued. The questions addressed in this work include the characterization of quaternionic Gabor systems that form frames, the characterization of those that are orthonormal bases, and the admissibility of such systems. We also explore necessary and/or sufficient conditions for Gabor frames. The issue of duality is also discussed. Furthermore, we study the stability of these systems.