Let G be a locally compact group and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the n-dimensional Fourier algebra, introduced by Todorov and Turowska, We investigate spectral synthesis properties of the multidimensional Fourier algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A^n(G).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In particular, we prove versions of the subgroup lemma, injection, and inverse projection theorems for both spectral sets and Ditkin sets. Additionally, we provide a result on the parallel synthesis between <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A^{n+1}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and finally prove Malliavin’s theorem.

Spectral Synthesis in Multidimensional Fourier Algebras

期刊论文
发表日期：

2026年8月5日

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Kanupriya,
N. Shravan Kumar

摘要

Let G be a locally compact group and let $A^n(G)$ $A^{n} (G)$ denote the n-dimensional Fourier algebra, introduced by Todorov and Turowska, We investigate spectral synthesis properties of the multidimensional Fourier algebra $A^n(G).$ $A^{n} (G) .$ In particular, we prove versions of the subgroup lemma, injection, and inverse projection theorems for both spectral sets and Ditkin sets. Additionally, we provide a result on the parallel synthesis between $A^n(G)$ $A^{n} (G)$ and $A^{n+1}(G)$ $A^{n + 1} (G)$ and finally prove Malliavin’s theorem.