<p>We extend the Kahane–Katznelson–de Leeuw theorem to smoothness spaces by showing that for any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g\in W^{l,2}(\mathbb{T}^{d})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>g</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mi>l</mi> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, there exists a function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f \in C^{l}(\mathbb{T}^{d})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>f</mi> <mo>∈</mo> <msup> <mi>C</mi> <mrow> <mi>l</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|\widehat{f}(n)|\ge|\widehat{g}(n)|$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> and <Equation ID="Equa"> <EquationSource Format="TEX">$\omega_{r}(D^{l}f,t)_{\infty}\approx \omega_{r}(D^{l}g,t)_{2},\qquad t >0.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>ω</mi> <mrow> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow> <mi>l</mi> </mrow> </msup> <mi>f</mi> <mo>,</mo> <mi>t</mi> <msub> <mo stretchy="false">)</mo> <mrow> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>≈</mo> <msub> <mi>ω</mi> <mrow> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow> <mi>l</mi> </mrow> </msup> <mi>g</mi> <mo>,</mo> <mi>t</mi> <msub> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <mi>t</mi> <mo>></mo> <mn>0.</mn> </math></EquationSource> </Equation></p><p>We apply this result to solve the Bernstein problem of finding necessary and sufficient conditions for the absolute convergence of multiple Fourier series. Finally, we explore the absolute integrability of Fourier transforms.</p>

Kahane–Katznelson–de Leeuw theorem and absolute convergence of Fourier series

期刊论文
发表日期：

2025年12月15日

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Miquel Saucedo,
Sergey Tikhonov

摘要

We extend the Kahane–Katznelson–de Leeuw theorem to smoothness spaces by showing that for any $g\in W^{l,2}(\mathbb{T}^{d})$ $g \in W^{l, 2} (T^{d})$ , there exists a function $f \in C^{l}(\mathbb{T}^{d})$ $f \in C^{l} (T^{d})$ satisfying $|\widehat{f}(n)|\ge|\widehat{g}(n)|$ $| \hat{f} (n) | \geq | \hat{g} (n) |$ and $\omega_{r}(D^{l}f,t)_{\infty}\approx \omega_{r}(D^{l}g,t)_{2},\qquad t >0.$ $ω_{r} (D^{l} f, t)_{\infty} \approx ω_{r} (D^{l} g, t)_{2}, t > 0.$

We apply this result to solve the Bernstein problem of finding necessary and sufficient conditions for the absolute convergence of multiple Fourier series. Finally, we explore the absolute integrability of Fourier transforms.