<p>The <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm of kernel functions plays a central role in studying the convergence properties of the Walsh-Fourier system. Although when viewed as a sequence, in many cases only their boundedness matters, the precise values of the individual terms can also be of interest. It is known that for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N \in \mathbb{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norms of the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>-th and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(2^{N+1} - 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-th kernel functions are exactly 1. In this paper, we prove that these are the only such cases: for every other <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n \in \mathbb{P}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">P</mi> </mrow> </math></EquationSource> </InlineEquation>, the strict inequality <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\|K_n\|_1 > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>K</mi> <mi>n</mi> </msub> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mn>1</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> holds. Finally, we present additional closed-form expressions for selected subsequences of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\|K_n\|_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>K</mi> <mi>n</mi> </msub> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> as illustrative examples, using the recursive formula of Toledo [8].</p>

Some notes on the $L^{1}$-norm of Walsh-Fejér kernels

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发表日期：

2026年8月3日

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I. Blahota

摘要

The $L^1$ $L^{1}$ -norm of kernel functions plays a central role in studying the convergence properties of the Walsh-Fourier system. Although when viewed as a sequence, in many cases only their boundedness matters, the precise values of the individual terms can also be of interest. It is known that for all $N \in \mathbb{N}$ $N \in N$ , the $L^1$ $L^{1}$ -norms of the $2^N$ $2^{N}$ -th and $(2^{N+1} - 1)$ $(2^{N + 1} - 1)$ -th kernel functions are exactly 1. In this paper, we prove that these are the only such cases: for every other $n \in \mathbb{P}$ $n \in P$ , the strict inequality $\|K_n\|_1 > 1$ $‖ K_{n} ‖_{1} > 1$ holds. Finally, we present additional closed-form expressions for selected subsequences of $\|K_n\|_1$ $‖ K_{n} ‖_{1}$ as illustrative examples, using the recursive formula of Toledo [8].

Some notes on the \(L^{1}\)-norm of Walsh-Fejér kernels

摘要