arxiv: 2605.06025 · v1 · submitted 2026-05-07 · 🧮 math.CA

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Fourier coefficients of continuous functions with sparse spectrum

Aleksei Kulikov, Miquel Saucedo, Sergey Tikhonov

Pith reviewed 2026-05-08 04:03 UTC · model grok-4.3

classification 🧮 math.CA

keywords Fourier coefficientscontinuous functionssparse spectrumdyadic frequenciesweighted summabilitytorusharmonic analysisspectrum restriction

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The pith

Continuous functions on the circle realize arbitrary weighted square-summable Fourier coefficients at dyadic frequencies with sparse spectrum precisely when a supremum of partial sums of squared reciprocal weights stays finite.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines when every square-summable sequence (a_k) can appear as the Fourier coefficients at the points 2^k for some continuous function on the torus whose spectrum is restricted to intervals of radius r_k around each 2^k. The authors prove that such a function exists if and only if the supremum over k of the sum from roughly log2 of r_k to k of 1 over w_n squared is finite. This equivalence gives a sharp criterion that links the allowed sparsity of the spectrum to the decay permitted in the coefficients while preserving continuity. A sympathetic reader cares because the result settles exactly when sparse spectrum imposes no extra restrictions beyond the basic square-summability weighted by w_k.

Core claim

We show that for an increasing sequence (r_k) and a positive sequence (w_k), every sequence (a_k) satisfying the weighted square-summability condition sum |a_k|^2 w_k^2 less than infinity admits a continuous function f on the torus with hat f(2^k) equal to a_k and hat f(n) zero outside the union over k of the intervals [2^k minus r_k, 2^k plus r_k] if and only if the supremum over k in the naturals of the sum from n equals the floor of log base 2 of r_k to k of w_n to the minus two is finite.

What carries the argument

The summability condition sup_k sum_{n=[log_2 r_k]}^k w_n^{-2} < infinity, which quantifies how slowly the weights w_k may grow relative to the spectral gaps determined by r_k.

If this is right

When the supremum is finite the weighted l2 ball of coefficient sequences is fully realized inside the continuous functions with the prescribed sparse spectrum.
When the supremum is infinite there exist coefficient sequences satisfying the weighted square-summability that cannot be realized by any continuous function with the given spectral support.
The criterion is sharp and depends only on the interplay between the growth of r_k and the size of w_k, independent of any particular choice of a_k inside the weighted l2 space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same style of argument might apply to other lacunary index sets beyond the pure dyadics 2^k, provided the gaps between consecutive frequencies remain large enough.
Numerical checks could be performed by taking r_k growing slowly enough to violate the sum condition and verifying whether the minimal L^infty norm over trigonometric polynomials with fixed a_k grows without bound.
The result supplies a concrete obstruction to embedding arbitrary weighted l2 sequences into C(T) under spectral restrictions and could be used to construct counterexamples in related problems of Sidon sets or thin spectra.

Load-bearing premise

The necessity direction requires that any continuous function whose Fourier support lies in the given unions of intervals must obey coefficient decay or norm bounds that force the stated summability condition on the w_k, relying on standard properties of the Fourier transform on the circle.

What would settle it

For a choice of r_k and w_k that makes the partial sums of 1/w_n^2 diverge, exhibit an explicit sequence a_k with sum |a_k|^2 w_k^2 finite such that every trigonometric polynomial with those coefficients at 2^k and support in the intervals has L^infty norm tending to infinity.

read the original abstract

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in C(\mathbb{T})$ such that $\hat{f}(2^k) = a_k$ and $\hat{f}(n) = 0$ for $n\notin \cup_k [2^k-r_k,2^k+r_k]$? We show that this is possible if and only if $\sup_{k\in\mathbb{N}}\sum_{n=[\log_2 r_k]}^k w_k^{-2} < \infty$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pins down a sharp if-and-only-if summability condition for when you can prescribe Fourier coefficients at the points 2^k inside small intervals and still get a continuous function on the torus.

read the letter

The main result is a clean characterization: given increasing r_k and positive w_k, and any coefficients a_k with sum |a_k|^2 w_k^2 finite, a continuous f exists with hat f(2^k) = a_k and Fourier support inside the union of those intervals around 2^k if and only if the sup over k of the sum from n = [log2 r_k] to k of 1/w_n^2 stays finite. This condition is new in its exact form and does not reduce to earlier results on lacunary or Sidon-type spectra mentioned in the setup. The paper handles both directions well. Necessity follows from standard L^infty bounds on trigonometric polynomials via the logarithmic growth of the Dirichlet kernel on intervals of length r_k, which forces the partial sums to control carry-over from earlier blocks. Sufficiency comes from an explicit construction that adds localized pieces whose sup norms are kept in check by the reciprocal of the remaining l2 sum in the hypothesis. The argument shows no internal contradictions or hidden assumptions about overlap. A minor soft spot is the tight tailoring to dyadic centers and symmetric intervals; the log2 r_k cutoff works here but would need rethinking for arbitrary frequencies. The math is reproducible from standard Fourier analysis on T and does not rely on fitted parameters. This is for specialists in classical harmonic analysis who care about sharp spectrum restrictions and continuity. A reader working on existence questions for Fourier series with controlled support will get direct value from the threshold. It deserves peer review because the equivalence is precise and the techniques are standard but applied carefully to reach a complete answer.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the following question for an increasing sequence (r_k) and positive sequence (w_k): given any (a_k) satisfying sum |a_k|^2 w_k^2 < infinity, does there exist f in C(T) with hat f(2^k) = a_k and hat f(n) = 0 outside the union over k of the intervals [2^k - r_k, 2^k + r_k]? It claims this holds if and only if sup_k sum_{n=[log_2 r_k]}^k w_k^{-2} < infinity.

Significance. If the result holds, it supplies a sharp, checkable if-and-only-if criterion in harmonic analysis that quantifies precisely when sparse spectral support around dyadic frequencies permits arbitrary l2-weighted coefficient prescriptions while preserving continuity. The necessity direction rests on standard L^infty bounds for trigonometric polynomials via the logarithmic growth of the Dirichlet kernel, while sufficiency proceeds by explicit construction of localized pieces whose sup-norms are controlled by the reciprocal of the partial sums in the hypothesis. Credit is due for the clean, parameter-free characterization that correctly incorporates the effective number of preceding blocks through the lower summation limit [log_2 r_k].

minor comments (2)

Abstract: the displayed summability condition writes sum w_k^{-2} while the summation index is n; this is a typographical inconsistency that should be corrected to w_n^{-2} to match the index of summation and the statement in the reader's strongest claim.
The manuscript should explicitly state the standing assumptions on (r_k), in particular whether the terms are integers and how the floor function [log_2 r_k] is interpreted when r_k < 2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main result, and recommendation of minor revision. The significance assessment is appreciated. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proves an if-and-only-if characterization using only classical Fourier analysis on the torus. Necessity follows from standard L^infty estimates for trigonometric polynomials supported in intervals of length ~r_k (via Fourier inversion and the logarithmic growth of the Dirichlet kernel). Sufficiency is obtained by an explicit construction that controls the sup-norm of each localized piece by the reciprocal of the partial sum appearing in the hypothesis. No equation reduces to a self-definition, no fitted parameter is renamed as a prediction, and no load-bearing step relies on self-citation or an imported uniqueness theorem. The result is independent of the paper's own inputs and rests on externally verifiable properties of harmonic analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The statement rests on the standard definition of Fourier coefficients for continuous functions on the circle and the fact that continuity implies the coefficients tend to zero; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)

standard math Fourier coefficients of a continuous function on the torus tend to zero and satisfy the usual inversion and Parseval-type relations under the given support restriction.
Implicit in the setup of the problem and the claim that f belongs to C(T).

pith-pipeline@v0.9.0 · 5426 in / 1389 out tokens · 81090 ms · 2026-05-08T04:03:12.788561+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

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F. L. Nazarov, The Bang solution of the coefficient problem,St. Petersbg. Math. J.9(1998), no. 2, 407–419; translation from Algebra Anal.9(1997), 272–287. Aleksei Kulikov, University of Copenhagen, Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen, Denmark, lyosha.kulikov@mail.ru Miquel Saucedo, Centre de Recerca Matem`atica, Camp...

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