Sharp $L^1$ Bounds for Approximate Convolution Idempotents

Definitions and mathematical statement

For $g\in L^1({\mathbb{R}})$ define convolution and the idempotency defect by \[ (g*g)(x)=\int_{\mathbb{R}} g(y)g(x-y)\,dy, \qquad J(g)=\|g*g-g\|_{L^1({\mathbb{R}})}. \] Young's inequality makes this a finite quantity. The source asks for its infimum among symmetric functions with $\|g\|_1=1$, and whether the infimum is attained.

Under the even-function reading, and for a specified scalar field $\mathbb K={\mathbb{R}}$ or ${\mathbb{C}}$, this is the problem \[ c_{\mathbb K}=\inf\{J(g):g\in L^1({\mathbb{R}};\mathbb K),\ g(-x)=g(x)\text{ a.e.},\ \|g\|_1=1\}, \] and whether there exists an admissible $g_*$ with $J(g_*)=c_{\mathbb K}$. The source does not specify real versus complex values or whether symmetric instead means the involutive condition $g(-x)=\overline{g(x)}$. It also does not impose $g\geq0$. Unit $L^1$ norm alone is not the condition $\int g=1$ for signed or complex functions. A probability-density version would be a different explicitly restricted class and is not silently imposed.

The triage checked 21 August 2026 reports the original lower bound and a numerical Gaussian candidate, with no proof of the exact infimum or attainment located. Neither the bound's numerical value nor the candidate formula is supplied here. A positive even Gaussian gives only a test in any larger signed/complex admissible class, not by itself a global minimizer theorem. Constant spatial dilations $g(x)\mapsto a g(ax)$ preserve both $\|g\|_1$ and $J(g)$, so scale choices of such a candidate would also need a consistent interpretation. This does not settle existence of any minimizer.

The requested outputs remain an exact value, or sharp characterization, of the infimum and a separate proof of attainment or nonattainment in the intended class. The dated failure to locate a proof is recorded as supplied literature triage, not a new status verification.

Short English statement

The question minimizes a convolution defect on a unit $L^1$ sphere and separately asks about attainment. The intended symmetry and whether signed or complex functions are allowed must be fixed. The supplied lower-bound and numerical Gaussian information gives no proven sharp value or minimizer.

Sources

• Local UnsolvedMath registry, permanent ID 30000041, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026.
• Mini-Workshop: Wavelets and Frames (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/10, pp. 479--544: https://doi.org/10.4171/owr/2004/10.
• Supplied literature reference "OWR 10/2004": https://doi.org/10.4171/OWR/2004/10.
• Supplied "Feichtinger survey": https://doi.org/10.1007/s43670-024-00095-9.
• Literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. No numerical bound or full candidate data are supplied.