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<p>In relation to Fuglede’s conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> that are in duality. In the terminology commonly used in the context of Fuglede’s conjecture, our result states that an open set tiles <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> by the finite set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\{0,1,\dots ,p-1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if it admits a spectrum (or, equivalently, a dual pair measure) given by the Lebesgue measure on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\left[ -\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb {Z}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="]" open="["> <mo>-</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </mstyle> <mo>,</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </mstyle> </mfenced> <mo>+</mo> <mi mathvariant="double-struck">Z</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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SOME PLANCHEREL IDENTITIES FOR UNBOUNDED SUBSETS OF \(\mathbb {R}\) IN DUALITY

  • 期刊论文

  • 发表日期：

    2026年3月28日

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  • Piyali Chakraborty,
  • Dorin Ervin Dutkay

摘要

In relation to Fuglede’s conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of \(\mathbb {R}\) R that are in duality. In the terminology commonly used in the context of Fuglede’s conjecture, our result states that an open set tiles \(\mathbb {R}\) R by the finite set \(\{0,1,\dots ,p-1\}\) { 0 , 1 , ⋯ , p - 1 } if and only if it admits a spectrum (or, equivalently, a dual pair measure) given by the Lebesgue measure on \(\left[ -\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb {Z}.\) - 1 2 p , 1 2 p + Z .