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4951 bytes, as of the version from 2026-10-25 18:58. Recorded here, not run.
# Build evidence for T499
Quantity: the Lebesgue constants of the degree-n Fourier projection on
continuous 2*pi-periodic functions, indexed by every integer 0 <= n <= 100.
L_0 is exactly 1. The other entries contain 100 significant decimal digits,
with proven ball bounds. This is one quantity, not the interpolation-node
Lebesgue constants stored in T328.
## Reproduction
Install numberdb in Sage, download the attached files into one directory, and
run `sage -python checks.py`. It reads the three small OEIS b-files locally;
there is no network dependency. Run `sage -python generate.py` to check all
stored entries against the generator. Reading the unpublished draft needs a
NumberDB API key configured in the standard client environment.
Download the authenticated JSON response from `/api/table?id=T499` and run
`sage -python checks.py stored-table.json` to test the written decimal
intervals against the independent computation. This checks the whole claimed
interval, rather than only its midpoint or a short decimal prefix.
## Independent integral computation
For 0 < t < pi, sin(t/2) is positive. The numerator sin((n+1/2)*t) has its
interior zeros at 2*j*pi/(2*n+1), j=1,...,n. Its sign starts positive and
alternates at those simple zeros. The primitive of
1+2*sum(cos(k*t), k=1,...,n) is t+2*sum(sin(k*t)/k, k=1,...,n).
The independent checker evaluates the differences of that primitive, with
alternating signs, then divides by pi. The normalized endpoints at 0 and pi
are exactly 0 and 1. It uses 640-bit balls and no tangent evaluations.
The controls L_0=1 and L_1=1/3+2*sqrt(3)/pi run before the table-wide checks.
Both sides of every comparison must be finite and have radius < 10^-110.
All 101 integral balls overlap the generator balls, and the entire integral
ball fits inside the serialized decimal's plus/minus-one-last-place interval.
Recomputing the generator at 589 rather than 429 bits also agrees throughout.
The uniform operator norm follows from the convolution representation:
the triangle inequality gives the bound by the normalized integral of |D_n|.
Continuous periodic approximations to sign(D_n(-t)) approach this bound at
x=0. The kernel has finitely many zeros, so smoothing the sign near its zeros
loses an arbitrarily small amount of the integral. Thus this is the sharp
norm, not only an upper bound.
## External evidence
- DLMF, equation 1.8.8, fixes the integral and n=0 indexing:
https://dlmf.nist.gov/1.8.E8
- I. A. Shakirov, On two-sided estimate for norm of Fourier operator,
Ufa Mathematical Journal 10(1) (2018), 94–114,
doi:10.13108/2018-10-1-94. The primary PDF was read using the web tool;
equations (3) and (5) give the integral and finite tangent sum.
A direct download from the build machine timed out, so no local PDF is
claimed as evidence. The formula and bibliographic data were accessible
through the web tool.
- Finch, Mathematical Constants (2003), section 4.2, pp. 250–255, is the
family reference. Its location was corroborated by the source bibliography;
this run does not claim to have read those book pages.
- MathWorld: https://mathworld.wolfram.com/LebesgueConstants.html
supplies the small-degree forms, including the independent radical for L_2.
- OEIS A226654, A226655, A226656: the attached b-files were fetched on
2026-10-25 from https://oeis.org/Axxxxxx/bxxxxxx.txt. They supply 104,
100 and 100 significant digits respectively. The checker verifies that
the whole ball falls within each decimal truncation cell, and uses the
shared agrees_with helper for the resulting exact integer prefixes.
## Decisions and scope
The proposal settles the normalization, indexing, rigour and precision.
This build chooses the consecutive range 0..100 as a compact reference to
initial Fourier truncation degrees. A measurement also evaluated n=500;
storage would permit more, but there was no reason to add that long tail.
The maximum radius is less than 6e-127 and the minimum relative accuracy is
420 bits at a working precision of 429 bits. Serialized values have length
101 characters, apart from the exact L_0, and the measured block is 13.5 KB.
The existing tags analysis, special values and Fourier series are used.
The family-proposed approximation theory tag does not exist. It could group
this table with T328 (interpolation Lebesgue constants), T111 (Bernstein
basis polynomials), and the other planned sharp-constant tables. This is a
tag proposal for a person, not a new singleton tag made by this build.
The corpus screening found no Fourier Lebesgue family. Its neighboring
interpolation table T328 and the Gibbs table T40 were read through the API,
and their actual URL slugs were used. No repeats declaration is appropriate:
the occasional numerical coincidence is not the same construction.
The build leaves the draft unpublished and not offered for review. Critique,
repair, and any offer are separate controller stages.