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Repeated integration by parts gives $∣ b_{n} (F) ∣ \leq 2 L^{5} M_{5} / (π^{5} n^{5})$ for odd $2 L$ -periodic $C^{5}$ functions and $∣ b_{n} (G) ∣ \leq 2 L^{4} M_{4} / (π^{4} n^{4})$ for $C^{4}$

Statement

Let $L > 0$ . Let $F : R \to R$ be odd and $2 L$ -periodic of class $C^{5}$ and let $G : R \to R$ be odd and $2 L$ -periodic of class $C^{4}$ (Sine coefficients, sine series and the odd periodic extension, $C^{k}$ Euclidean maps and diffeomorphisms); put

$M_{5} : = \sup_{R} ∣ F^{(5)} ∣ < + \infty, M_{4} : = \sup_{R} ∣ G^{(4)} ∣ < + \infty .$

Then for every integer $n \geq 1$ , with the sine coefficients $b_{n}$ of Sine coefficients, sine series and the odd periodic extension,

$∣ b_{n} (F) ∣ \leq \frac{2 L^{5} M_{5}}{π^{5} n^{5}}, ∣ b_{n} (G) ∣ \leq \frac{2 L^{4} M_{4}}{π^{4} n^{4}} .$

Facts & Assumptions

Given: The period $L > 0$ , the functions $F, G$ and the bounds $M_{5}, M_{4}$ ; an integer $n \geq 1$ .

[F1]

Integration by parts: for $C^{1}$ functions $p, q$ on $[0, L]$ , their derivatives and products are continuous and hence integrable there; therefore $\int_{0}^{L} p q^{'} = [p q]_{0}^{L} - \int_{0}^{L} p^{'} q$ (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ , For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F2]

Boundary values: an odd $2 L$ -periodic function $W$ satisfies $W (0) = 0$ and $W (L) = W (- L) = - W (L)$ , hence $W (L) = 0$ ; its even-order derivatives are again odd $2 L$ -periodic, so $W^{''} (0) = W^{''} (L) = W^{''''} (0) = W^{''''} (L) = 0$ whenever these derivatives exist. Also $\sin (n π \cdot 0 / L) = \sin 0 = 0$ and $\sin (n π L / L) = \sin (n π) = 0$ , while $\cos (0) = 1$ and $\cos (n π) = (- 1)^{n}$ (Sine and cosine defined by their real power series, The derivatives of sine and cosine are cosine and minus sine, The zero sets of sine and cosine and the least positive common period 2 pi, Sine coefficients, sine series and the odd periodic extension).

[F3]

The bounds $M_{5}, M_{4}$ dominate the corresponding derivatives everywhere, so in particular on $[0, L]$ ; and $∣ b_{n} ∣ = ∣ \frac{2}{L} \int_{0}^{L} F \sin (n π x / L) d x ∣ \leq \frac{2}{L} \int_{0}^{L} ∣ F ∣$ when used with the sup bound (Absolute value in an ordered field, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Open ball, closed ball and sphere in a metric space, Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ).

Proof

technique · direct; integrate by parts five times (four times for $G$) and use the vanishing boundary values of the odd derivatives

1.1givenF1F2algebra

Put $I_{0} : = \int_{0}^{L} F (x) \sin (n π x / L) d x$ and $I_{j} : = \int_{0}^{L} F^{(j)} (x) \cdot (\sin if j even, \cos if j odd) (n π x / L) d x$ for $j = 1, \dots, 5$ . Applying [F1] successively, each integration by parts produces the boundary term $[F^{(j)} (x) \cdot (\pm \frac{L}{n π}) (the other trigonometric factor)]_{0}^{L},$ which vanishes by [F2]: at $j$ even the boundary factor is $\cos$ , nonzero at the endpoints, but the multiplying derivative $F^{(j)}$ is odd, hence vanishes at $0$ and $L$ ; at $j$ odd the boundary factor is $\sin$ , which vanishes at $0$ and $L$ . Therefore $I_{0} = \frac{L}{n π} I_{1}$ , $I_{1} = - \frac{L}{n π} I_{2}$ , $I_{2} = \frac{L}{n π} I_{3}$ , $I_{3} = - \frac{L}{n π} I_{4}$ , $I_{4} = \frac{L}{n π} I_{5}$ , and chaining, $I_{0} = (\frac{L}{n π})^{5} I_{5} .$

2.1step 1.1F3algebra∎

Since $∣ I_{5} ∣ \leq \int_{0}^{L} ∣ F^{(5)} ∣ d x \leq L M_{5}$ , step 1.1 gives $∣ b_{n} (F) ∣ = \frac{2}{L} ∣ I_{0} ∣ \leq \frac{2}{L} (\frac{L}{n π})^{5} L M_{5} = \frac{2 L^{5} M_{5}}{π^{5} n^{5}}$ . The same computation with four integrations by parts and $G^{(4)}$ in place of $F^{(5)}$ gives $∣ b_{n} (G) ∣ \leq \frac{2 L^{4} M_{4}}{π^{4} n^{4}}$ .

Remarks

The powers $n^{- 5}$ and $n^{- 4}$ are exactly the number of integrations by parts available; from $C^{k}$ odd periodic functions one obtains $O (n^{- k})$ (with the parity pattern of the boundary terms as used above). No sharpness is claimed.
The estimates are uniform in $n$ , and their summands are summable; this is what permits the termwise differentiation of the string series in the companion theorem. The constant depends on $L$ , on the sup norms and on the fixed exponent, not on $n$ .

Repeated integration by parts gives $∣ b_{n} (F) ∣ \leq 2 L^{5} M_{5} / (π^{5} n^{5})$ for odd $2 L$ -periodic $C^{5}$ functions and $∣ b_{n} (G) ∣ \leq 2 L^{4} M_{4} / (π^{4} n^{4})$ for $C^{4}$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Repeated integration by parts gives ∣bn(F)∣≤2L5M5/(π5n5)∣bn​(F)∣≤2L5M5​/(π5n5) for odd 2L2L-periodic C5C5 functions and ∣bn(G)∣≤2L4M4/(π4n4)∣bn​(G)∣≤2L4M4​/(π4n4) for C4C4

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Repeated integration by parts gives $∣ b_{n} (F) ∣ \leq 2 L^{5} M_{5} / (π^{5} n^{5})$ for odd $2 L$ -periodic $C^{5}$ functions and $∣ b_{n} (G) ∣ \leq 2 L^{4} M_{4} / (π^{4} n^{4})$ for $C^{4}$