We consider functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f: \mathbb {Z}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and kernels <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u: \{-n, \cdots , n\} \rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>:</mo> <mo stretchy="false">{</mo> <mo>-</mo> <mi>n</mi> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> normalized by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sum _{\ell = -n}^{n} u(\ell ) = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>-</mo> <mi>n</mi> </mrow> <mi>n</mi> </msubsup> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, making the convolution <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mrow /> <mo>∗</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> a “smoother” local average of f. We identify which choice of u most effectively smooths the second derivative in the following sense. For each u, basic Fourier analysis implies there is a constant C(u) so <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Vert \Delta (u f)\Vert _{\ell ^2(\mathbb {Z})} \le C(u)\Vert f\Vert _{\ell ^2(\mathbb {Z})}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mrow /> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≤</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </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">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f: \mathbb {Z}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we find an explicit expression for the unique kernel u that minimizes C(u). The minimizing kernel is remarkably close to the Epanechnikov kernel in Statistics. This solves a problem of Kravitz-Steinerberger and an extremal problem for polynomials is solved as a byproduct.

A Sharp Fourier Inequality and the Epanechnikov Kernel

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发表日期：

2026年1月8日

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Sean Richardson

摘要

We consider functions $f: \mathbb {Z}\rightarrow \mathbb {R}$ $f : Z \to R$ and kernels $u: \{-n, \cdots , n\} \rightarrow \mathbb {R}$ $u : {- n, \dots, n} \to R$ normalized by $\sum _{\ell = -n}^{n} u(\ell ) = 1$ $\sum_{ℓ = - n}^{n} u (ℓ) = 1$ , making the convolution $u *f$ $u * f$ a “smoother” local average of f. We identify which choice of u most effectively smooths the second derivative in the following sense. For each u, basic Fourier analysis implies there is a constant C(u) so $\Vert \Delta (u *f)\Vert _{\ell ^2(\mathbb {Z})} \le C(u)\Vert f\Vert _{\ell ^2(\mathbb {Z})}$ ${‖ Δ (u * f) ‖}_{ℓ^{2} (Z)} \leq C (u) {‖ f ‖}_{ℓ^{2} (Z)}$ for all $f: \mathbb {Z}\rightarrow \mathbb {R}$ $f : Z \to R$ . In this paper, we find an explicit expression for the unique kernel u that minimizes C(u). The minimizing kernel is remarkably close to the Epanechnikov kernel in Statistics. This solves a problem of Kravitz-Steinerberger and an extremal problem for polynomials is solved as a byproduct.