For a function F represented as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F(x)=\sum _{n=0}^\infty {f_n (x) e^{2 \pi i \lambda _n x}},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mi>x</mi> </mrow> </msup> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where each <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\operatorname {spec}(f_n) \subset [0, 1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>spec</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\lambda _n)_{n\ge 0}\subset \mathbb {R}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation> is a lacunary sequence, we obtain <Equation ID="Equ7"> <EquationSource Format="TEX">$ \Vert F\Vert _{L^2(\mathbb {R})}\lesssim \Vert F\chi _{E}\Vert _{L^2(\mathbb {R})} $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>F</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≲</mo> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>F</mi> <msub> <mi>χ</mi> <mi>E</mi> </msub> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </mrow> </math></EquationSource> </Equation>provided that E is a thick subset of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. This extends the Logvinenko-Sereda theorem and answers a question posed by Kovrizhkin for functions with positive frequencies.

A Logvinenko-Sereda Theorem for Lacunary Spectra

期刊论文
发表日期：

2026年8月3日

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Miquel Saucedo,
Sergey Tikhonov

摘要

For a function F represented as $F(x)=\sum _{n=0}^\infty {f_n (x) e^{2 \pi i \lambda _n x}},$ $F (x) = \sum_{n = 0}^{\infty} f_{n} (x) e^{2 π i λ_{n} x},$ where each $f_n$ $f_{n}$ satisfies $\operatorname {spec}(f_n) \subset [0, 1]$ $spec (f_{n}) \subset [0, 1]$ and $(\lambda _n)_{n\ge 0}\subset \mathbb {R}_+$ ${(λ_{n})}_{n \geq 0} \subset R_{+}$ is a lacunary sequence, we obtain $ \Vert F\Vert _{L^2(\mathbb {R})}\lesssim \Vert F\chi _{E}\Vert _{L^2(\mathbb {R})} $ ${‖ F ‖}_{L^{2} (R)} ≲ {‖ F χ_{E} ‖}_{L^{2} (R)}$ provided that E is a thick subset of $\mathbb {R}$ $R$ . This extends the Logvinenko-Sereda theorem and answers a question posed by Kovrizhkin for functions with positive frequencies.