<p>Due to Walsh functions being a discontinuous alternative to trigonometric functions, in contrast to usual trigonometric function-based Gabor analysis, the Walsh function-based Gabor analysis has potential applications in processing discontinuous signals. This paper addresses such Gabor analysis in the setting of periodic subspace <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{2}(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S\subseteq \mathbb R_{+}=[0,\,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊆</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em" /> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S\oplus \alpha \mathbb Z_{+}=S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊕</mo> <mi>α</mi> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> <mo>=</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and “<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\oplus $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⊕</mo> </math></EquationSource> </InlineEquation>" is a kind of Walsh addition defined on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb R_{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation>. Using “<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\oplus $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⊕</mo> </math></EquationSource> </InlineEquation>"-based Zak transform, we characterize the complete condition of Gabor systems, Gabor frames (Riesz bases, orthonormal bases) and (weak) Gabor duals in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^{2}(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Also some examples have been provided.</p>

Subspace Gabor frames and their (weak) Gabor duals on periodic subsets of the half real line

期刊论文
发表日期：

2026年3月12日

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Yan Zhang,
Yun-Zhang Li

摘要

Due to Walsh functions being a discontinuous alternative to trigonometric functions, in contrast to usual trigonometric function-based Gabor analysis, the Walsh function-based Gabor analysis has potential applications in processing discontinuous signals. This paper addresses such Gabor analysis in the setting of periodic subspace $L^{2}(S)$ $L^{2} (S)$ with $S\subseteq \mathbb R_{+}=[0,\,\infty )$ $S \subseteq R_{+} = [0, \infty)$ and $S\oplus \alpha \mathbb Z_{+}=S$ $S \oplus α Z_{+} = S$ , where $\alpha >0$ $α > 0$ and “ $\oplus $ $\oplus$ " is a kind of Walsh addition defined on $\mathbb R_{+}$ $R_{+}$ . Using “ $\oplus $ $\oplus$ "-based Zak transform, we characterize the complete condition of Gabor systems, Gabor frames (Riesz bases, orthonormal bases) and (weak) Gabor duals in $L^{2}(S)$ $L^{2} (S)$ . Also some examples have been provided.