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FALSE: the $O (N^{- 4})$ sine-tail bound holds for every continuous odd periodic function. A $C^{1}$ datum with coefficients $n^{- 3}$ refutes it

Statement refuted

False statement. For every continuous odd $2 L$ -periodic function $F$ there is a constant $C_{F}$ such that, for every $N \geq 1$ ,

$\sup_{x \in R} ∣ F (x) - S_{N}^{s} F (x) ∣ \leq C_{F} (N + 1)^{- 4},$

where $S_{N}^{s} F$ is the $N$ -th partial sum of the sine series of Sine coefficients, sine series and the odd periodic extension.

(The theorem of this collection proves the bound only under the $C^{5}$ hypothesis of The classical $C^{2}$ fixed-end string: the sine series solves the wave equation with the given data, uniquely, with an explicit tail bound; the statement above drops that hypothesis, and that is exactly what fails.)

Facts & Assumptions

Given: A period $L > 0$ and the function $F : R \to R$ defined by the absolutely convergent series

$F (x) : = \sum_{n \geq 1} n^{- 3} \sin (\frac{n π x}{L}) .$

[L1]

Convergence and regularity: $\sum n^{- 3} < \infty$ and $\sum n^{- 2} < \infty$ , so by the Weierstrass majorant test the series for $F$ and for the formal derivative $\sum (π / L) n^{- 2} \cos (n π x / L)$ converge uniformly; hence $F$ is continuous, odd and $2 L$ -periodic; applying If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit to partial sums on each compact interval gives $C^{1}$ regularity with the displayed derivative (The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series, A series of real-valued functions and its pointwise and uniform convergence through its partial sums, The uniform limit of continuous real-valued functions on a metric space is continuous, $C^{k}$ Euclidean maps and diffeomorphisms).

[L2]

Coefficients: Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error permits termwise integration of the uniformly convergent series against $\sin (k π x / L)$ using the orthogonality $\int_{0}^{L} \sin (n π x / L) \sin (k π x / L) d x = (L / 2) δ_{n k}$ gives $b_{k} (F) = k^{- 3}$ for every $k \geq 1$ (Sine coefficients, sine series and the odd periodic extension, Sine orthogonality on a half period and the full-period integrals of odd $2 L$ -periodic functions, Integral over a measurable subset, Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ).

[L3]

For fixed $N$ , put $P_{N, m} : = \sum_{n = N + 1}^{m} n^{- 3} \sin (n π x / L)$ . Finite expansion and orthogonality give $\int_{0}^{L} P_{N, m}^{2} = \frac{L}{2} \sum_{n = N + 1}^{m} n^{- 6}$ . The tails $P_{N, m}$ converge uniformly to $R_{N}$ , are uniformly bounded, and therefore their squares converge uniformly too. The uniform integral-error bound yields $\int_{0}^{L} R_{N}^{2} = \frac{L}{2} \sum_{n > N} n^{- 6}$ (Sine orthogonality on a half period and the full-period integrals of odd $2 L$ -periodic functions, Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error).

[L4]

For every integer $M > N$ , the decreasing function $x \mapsto x^{- 6}$ satisfies $\sum_{n = N + 1}^{M} n^{- 6} \geq \int_{N + 1}^{M + 1} x^{- 6} d x = \frac{1}{5} ((N + 1)^{- 5} - (M + 1)^{- 5})$ : on each $[n, n + 1]$ , $x^{- 6} \leq n^{- 6}$ . The series converges by the integral test, so letting $M \to \infty$ gives $\sum_{n > N} n^{- 6} \geq \frac{1}{5} (N + 1)^{- 5}$ (The integral test: for $f \geq 0$ nonincreasing on $[0, \infty)$ , $\sum_{k} f (k)$ converges if and only if the sequence $(\int_{0}^{N} f)_{N}$ is bounded, with $\int_{0}^{N} f \leq \sum_{k < N} f (k) \leq f (0) + \int_{0}^{N} f$ , The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive).

Counterexample

technique · direct; the tail's $L^2$ norm forces its sup norm to exceed any $C(N+1)^{-4}$ along a subsequence

1.1givenL1L2L3L4algebra

By [L2] each $S_{N}^{s} F = \sum_{n \leq N} n^{- 3} \sin (n π x / L)$ is continuous, and $F$ is continuous by [L1]; hence each $R_{N}$ is continuous and $2 L$ -periodic. By [L3] and [L4], $\int_{0}^{L} R_{N} (x)^{2} d x = \frac{L}{2} \sum_{n > N} n^{- 6} \geq \frac{L}{10} (N + 1)^{- 5} .$

2.1step 1.1L1algebra

Since $R_{N}$ is continuous on the compact interval $[0, L]$ by [L1], its squared sup norm dominates its mean: $\sup_{x} R_{N} (x)^{2} \geq \frac{1}{L} \int_{0}^{L} R_{N}^{2} d x \geq \frac{1}{10} (N + 1)^{- 5}$ . Taking square roots, $\sup_{x \in R} ∣ F (x) - S_{N}^{s} F (x) ∣ \geq \frac{1}{\sqrt{10}} (N + 1)^{- 5 / 2} .$

3.1step 2.1algebra∎

Suppose a constant $C_{F}$ satisfied the false statement's bound for all $N$ . Then for every $N$ , $\frac{1}{\sqrt{10}} (N + 1)^{- 5 / 2} \leq C_{F} (N + 1)^{- 4}$ , i.e. $C_{F} \geq \frac{1}{\sqrt{10}} (N + 1)^{3 / 2}$ ; letting $N \to \infty$ gives $C_{F} \geq + \infty$ , a contradiction. Hence no such $C_{F}$ exists, and the false statement is refuted.

Remarks

The datum is $C^{1}$ and is not $C^{5}$ : otherwise the coefficient-decay lemma would bound $n^{- 3}$ by a fixed multiple of $n^{- 5}$ for every $n$ , which is impossible. Nonabsolute convergence of a formally differentiated series alone would not prove failure of $C^{2}$ regularity. The counterexample refutes the hypothesis-free estimate; it does not claim $C^{5}$ is necessary for every datum having an $O (N^{- 4})$ tail.
The lower bound uses the $L^{2}$ tail of the sine coefficients; it shows the tail cannot be better than order $N^{- 5 / 2}$ in sup norm for this datum, while any $O (N^{- 4})$ assertion would require the considerably faster decay of the $C^{5}$ class.

Depends on

Dependency tree · 2 levels

Sources

R. Laugesen, Harmonic Analysis Lecture Notes (arXiv:0903.3845), Chapters 1 and 5 (smoothness and decay of Fourier coefficients; Fourier series in L²(T))

FALSE: the O(N−4)O(N−4) sine-tail bound holds for every continuous odd periodic function. A C1C1 datum with coefficients n−3n−3 refutes it