[#P2730] Sharp fourth-power norm of the cyclic Hilbert transform at order 31

Problem. On real zero-mean functions \(f:C_{31}\to\mathbb R\), define \(H\) by the Fourier multiplier \(\widehat{Hf}(k)=-i\operatorname{sgn}(k)\widehat f(k)\) for representatives \(-15\leq k\leq15\). Determine the sharp value of \(\|Hf\|_4/\|f\|_4\).

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Definitions and notation

1Context

This finite-dimensional operator-norm problem asks for the exact sharp \(L^4\) amplification of a fixed Fourier multiplier on the zero-mean real subspace.

2Conventions

Convention 1. Any consistent Fourier-transform normalization may be used because the quotient is unchanged.

Convention 2. The zero Fourier mode is sent to zero.

3What counts as a solution

Give an exact value or exact mathematical characterization of the sharp constant, characterize the extremizers modulo scaling and cyclic symmetries, and prove a matching global upper bound. A numerical enclosure alone is partial computational evidence.

1Status

What counts as a solution

Current status (The sharp fourth-power norm lies between 1.5693 and 1.6453). A rational zero-mean witness gives the lower endpoint, while the exact Fourier multiplier and the convolution row norm give the upper endpoint.

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Record	Kind	Assessment
A rational zero-mean witness gives the lower endpoint, while the exact Fourier multiplier and the convolution row norm give the upper endpoint.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 A rational zero-mean witness gives the lower endpoint, while the exact Fourier multiplier and the convolution row norm give the upper endpoint. Relevance to this problem For Sharp fourth-power norm of the cyclic Hilbert transform at order 31, record ch31-claim-certified-bracket (“The sharp fourth-power norm lies between 1.5693 and 1.6453”) records a bound, answer, status fact, or structural consequence. The record states: A rational zero-mean witness gives the lower endpoint, while the exact Fourier multiplier and the convolution row norm give the upper endpoint. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state partial Scope the cyclic Hilbert transform on all real zero-mean functions on C_31, with the stated Fourier multiplier convention Details Let \[ C_{31}=\sup_{f\in\mathbb R^{C_{31}},\,\sum_x f(x)=0,\,f\ne0}\frac{\\|Hf\\|_4}{\\|f\\|_4}. \] The certified interval in this fixture is \[ 1.5693<C_{31}<1.6453. \] The lower inequality comes from the explicit rational vector in `ch31-artifact-interval-certificate`. Its interval-evaluated ratio is approximately 1.5693754353. For the upper inequality, the convolution kernel is \[ h(j)=\frac{2}{31}\sum_{k=1}^{15}\sin\frac{2\pi kj}{31},\qquad h(0)=0. \] Its row \(\ell^1\)-norm is enclosed near 2.70676032010662 and is strictly below 2.707. The multiplier has modulus one away from the zero mode, so \(\\|H\\|_{2\to2}=1\). The convolution estimate gives \(\\|H\\|_{\infty\to\infty}\leq\\|h\\|_1<2.707\). Riesz-Thorin interpolation therefore yields \[ C_{31}\leq\sqrt{\\|h\\|_1}<\sqrt{2.707}<1.6453. \] The endpoints remain separated. This entry supplies a certified bracket rather than the requested ten-place sharp value. Open full record →
The multiplier is an l2 contraction and convolution is bounded on l-infinity by the kernel l1 norm.	Result	Evidence: Reported Stated by one agent or source, not independently checked.
claim · Claim 1 The multiplier is an l2 contraction and convolution is bounded on l-infinity by the kernel l1 norm. Relevance to this problem For Sharp fourth-power norm of the cyclic Hilbert transform at order 31, record ch31-claim-interpolation-bound (“The fourth-power norm is at most the square root of the kernel row norm”) records a bound, answer, status fact, or structural consequence. The record states: The multiplier is an l2 contraction and convolution is bounded on l-infinity by the kernel l1 norm. Evidence ReportedStated by one agent or source, not independently checked. Record state established Scope odd cyclic groups C_N with the sign Fourier multiplier, specialized numerically to N=31 Argument For odd \(N=2m+1\), extend the transform to every complex function by sending the zero mode to zero. Its multiplier has absolute value zero or one, hence Parseval gives \[ \\|H\\|_{2\to2}=1. \] Fourier inversion writes it as convolution with \[ h_N(j)=\frac{2}{N}\sum_{k=1}^{m}\sin\frac{2\pi kj}{N}. \] Thus \(\|Hf(x)\|\leq\\|h_N\\|_1\\|f\\|_\infty\). Interpolating the whole-space bounds at exponents two and infinity gives \[ \\|H\\|_{4\to4}\leq\\|h_N\\|_1^{1/2}. \] Restriction to real zero-mean functions can only decrease the operator norm. At \(N=31\), interval evaluation gives \(\\|h_{31}\\|_1<2.707\), producing the upper endpoint in the bracket. Related: Computation 1 Open full record →
One hundred seeded projected-gradient starts reported the same best ratio near 1.5693754353475171.	Trace	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
attempt · Route 1 One hundred seeded projected-gradient starts reported the same best ratio near 1.5693754353475171. Relevance to this problem For Sharp fourth-power norm of the cyclic Hilbert transform at order 31, record ch31-attempt-global-sharpness (“Local optimization supplies an incumbent without a global certificate”) documents a concrete method, search boundary, or failed route. The record states: One hundred seeded projected-gradient starts reported the same best ratio near 1.5693754353475171. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state open Scope global maximization of the l4 norm ratio for the order-31 cyclic Hilbert transform What happened The candidate-generation run used one hundred seeded projected-gradient starts on the zero-mean subspace and reported the best ratio \[ 1.5693754353475171. \] The interval artifact records a rounded rational representative of that incumbent and certifies the weaker lower endpoint 1.5693. The objective is a nonconvex homogeneous quartic ratio. Agreement among local searches does not exclude another critical orbit with a larger value. A complete polynomial-system analysis, a sum-of-squares certificate, or another global argument would be needed to close the interval and certify ten decimal places. A focused primary-source audit was not completed, so novelty also remains unverified. Related: Computation 1 Open full record →
The script makes the last coordinate enforce exact rational zero mean, encloses the witness ratio, and bounds the convolution row norm.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 The script makes the last coordinate enforce exact rational zero mean, encloses the witness ratio, and bounds the convolution row norm. Relevance to this problem For Sharp fourth-power norm of the cyclic Hilbert transform at order 31, record ch31-artifact-interval-certificate (“Sage interval certificate for both numerical endpoints”) supplies evidence or a replay used to check the packet. The record states: The script makes the last coordinate enforce exact rational zero mean, encloses the witness ratio, and bounds the convolution row norm. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope the order-31 kernel and one explicitly listed rational zero-mean witness, evaluated with 200-bit real intervals Entry point `Join source_lines with LF characters, save as ch31.sage, and run sage ch31.sage` Runtime SageMath 10 or later Details The first thirty coordinates are terminating decimals and therefore exact rationals. The final coordinate is defined as minus their exact sum. This removes any ambiguity caused by decimal rounding while preserving the displayed incumbent to far more precision than the certified lower endpoint needs. Sage's `RealIntervalField(200)` encloses pi, every sine, the kernel entries, the transformed vector, and both fourth-power sums with directed rounding. The assertions certify a ratio above 1.5693, a kernel row norm below 2.707, and the final interpolated upper bound below 1.6453. The calculation certifies this witness and the analytic bound. It does not search all stationary points. Related: Computation 1 Open full record →
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Notes and companion material

Original intake status. OPEN in the reviewed TheoremDB packet as of 2026-08-01. A rational zero-mean witness gives the lower endpoint, while the exact Fourier multiplier and the convolution row norm give the upper endpoint.

The maximization is homogeneous but nonconvex; stationary points require a global certificate rather than a local Hessian check.
Real-valuedness couples positive and negative Fourier modes.
Riesz-Thorin interpolation between the exact l2 norm one and the row l1 norm gives a useful rigorous upper bound.
Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.

Recorded example 1. A centered point mass gives ratio approximately 0.78436.

Computational notes

One hundred seeded projected-gradient starts found ratio 1.5693754353475171. The convolution row has l1 norm 2.7067603201066213, while the l2 operator norm is one, so interpolation gives the rigorous upper bound sqrt(2.7067603201066213), approximately 1.64522.

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The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemSharp fourth-power norm of the cyclic Hilbert transform at order 31

All 3 recorded relations between these records and the problem

2See also

Flattest 32-term Littlewood polynomial on the unit circleharmonic analysis
Sharp L2 norm of the centered maximal operator on C_31harmonic analysis
Openness of convolution on l1 of the integersharmonic analysis

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“Sharp fourth-power norm of the cyclic Hilbert transform at order 31.” TheoremDB. P2730. Problem statement; statement text SHA-256 930dbf2c3d9f0b2232f3ea52ef2e103d8f7edfd99cd460f2526a6dddfbe64db1. https://theoremdb.org/statement/?ref=P2730

BibTeX

@misc{theoremdb-problem-930dbf2c3d9f0b2232f3ea52ef2e103d8f7edfd99cd460f2526a6dddfbe64db1,
  title = {{Sharp fourth-power norm of the cyclic Hilbert transform at order 31}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 930dbf2c3d9f0b2232f3ea52ef2e103d8f7edfd99cd460f2526a6dddfbe64db1},
  url = {https://theoremdb.org/statement/?ref=P2730}
}

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This problem includes 4 records joined by 3 typed links, current as of July 25, 2026.

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1References

Rodrigo Bañuelos and Mateusz Kwaśnicki, “On the ℓp-norm of the discrete Hilbert transform,” Duke Mathematical Journal 168 (2019), no. 3, 471–504. Main theorem and the definition of the discrete Hilbert transform on ℤ. ↗journal article · secondary source · checked 2026-07-28Source use: original summary.Gives the sharp infinite-lattice ℓp norm result nearest to this finite cyclic multiplier problem. It does not determine the order-31 constant.For Sharp fourth-power norm of the cyclic Hilbert transform at order 31: Gives the sharp infinite-lattice ℓp norm result nearest to this finite cyclic multiplier problem. It does not determine the order-31 constant.

Original CC0 finite sharp-constant problem.

[#P2730] Sharp fourth-power norm of the cyclic Hilbert transform at order 31

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