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<p>We study the short-time Fourier transform phase retrieval problem in locally compact abelian groups. Using probabilistic methods, we show that for a large class of groups <i>G</i> and compact subsets <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K\subseteq G\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⊆</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> there exists a window function and a uniformly discrete set in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(G\times \widehat{G}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>×</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> allowing phase retrieval in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(K).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We also study the obstructions for STFT phase retrieval on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^2(G),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> motivating the restriction to compactly supported function spaces.</p>

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Phase retrieval from short-time Fourier transform in LCA groups

  • 期刊论文

  • 发表日期：

    2026年1月24日

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  • Natalia Accomazzo,
  • Daniel Carando,
  • Rocío Nores,
  • Victoria Paternostro,
  • Sebastián Velazquez

摘要

We study the short-time Fourier transform phase retrieval problem in locally compact abelian groups. Using probabilistic methods, we show that for a large class of groups G and compact subsets \(K\subseteq G\) K ⊆ G there exists a window function and a uniformly discrete set in \(G\times \widehat{G}\) G × G ^ allowing phase retrieval in \(L^2(K).\) L 2 ( K ) . We also study the obstructions for STFT phase retrieval on \(L^2(G),\) L 2 ( G ) , motivating the restriction to compactly supported function spaces.