Skip to content
DraconiticResearch
Search Sources API
197,761 studies31 sourcesUpdated 35 min ago
PreprintarXiv licencearXiv

A convolution inequality and the Krzyż conjecture

  • Sullivan Francis MacDonald
Open at sourcePDFDOICite

Abstract

Using a sharp bound for convolutions of non-negative trigonometric polynomials with the Fejér kernel resembling a pointwise variant of Bernstein's inequality for polynomials, we prove new properties of extremal functions for the Krzyż conjecture. In particular, we resolve a 1993 conjecture of Peretz by showing that if f is extremal for some coefficient, then |f(0)|≤ 1/e. Furthermore, we show that if equality holds in this estimate for every extremal function, then the Krzyż conjecture is true.

Related

  1. Krzyż's conjecture for the seventh coefficient
    Xiaohui Xie
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. On extremal problems related to the Krzyż conjecture
    Sarah May Instanes, Adrián Llinares, Dragan Vukotić
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  3. A computer-assisted proof of Krzyż's conjecture for the sixth coefficient
    Frank Wikström
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Improved Brunn--Minkowski inequality for θ-convolution bodies
    Javier Martín Goñi
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  5. The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups
    Yong Jiao, Sijie Luo, Dmitriy Zanin +1
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{macdonald2026convolution,
  title = {{A convolution inequality and the Krzyż conjecture}},
  author = {Sullivan Francis MacDonald},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05363},
  eprint = {2611.05363},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05363},
}
RIS
TY  - UNPB
TI  - A convolution inequality and the Krzyż conjecture
AU  - Sullivan Francis MacDonald
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05363
UR  - https://arxiv.org/abs/2611.05363
AB  - Using a sharp bound for convolutions of non-negative trigonometric polynomials with the Fejér kernel resembling a pointwise variant of Bernstein's inequality for polynomials, we prove new properties of extremal functions for the Krzyż conjecture. In particular, we resolve a 1993 conjecture of Peretz by showing that if f is extremal for some coefficient, then |f(0)|≤ 1/e. Furthermore, we show that if equality holds in this estimate for every extremal function, then the Krzyż conjecture is true.
ER  -