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Fejer summation of the square wave

Example

Let $q$ be the one-periodic square wave defined by

$q (x) = {\begin{cases} 1, & 0 < x < 1 / 2, \\ - 1, & - 1 / 2 < x < 0, \end{cases}$

with $q (0) = q (1 / 2) = 0$ . Then

$σ_{N} q (x) = \frac{4}{π} \sum_{m = 0}^{⌊ (N - 1) / 2 ⌋} (1 - \frac{2 m + 1}{N + 1}) \frac{\sin (2 π (2 m + 1) x)}{2 m + 1},$

and

$σ_{N} q (0) \to 0.$

Moreover $∣ σ_{N} q (x) ∣ \leq 1$ for every $x$ and every $N$ , so these positive means recover the midpoint value without a fixed Gibbs overshoot.

Facts & Assumptions

Given: The one-periodic square wave $q$ above.

[L1]

The Cesaro means are defined by averaging Fourier partial sums (Cesaro and Abel means of a Fourier series).

[L2]

The Fejer kernels are nonnegative and have Riemann integral $1$ (The Fejer kernel is a positive approximate identity).

Verification

technique · direct

1.1L1algebra

Direct integration gives $\hat{q} (2 m) = 0$ for every $m$ and $\hat{q} (2 m + 1) = \frac{2}{π i (2 m + 1)}, \hat{q} (- (2 m + 1)) = - \frac{2}{π i (2 m + 1)} .$ Hence $S_{j} q (x) = \frac{4}{π} \sum_{0 \leq m, 2 m + 1 \leq j} \frac{\sin (2 π (2 m + 1) x)}{2 m + 1} .$

2.1L1step 1.1algebra

Averaging the partial sums from step 1.1 as in [L1] shows that the odd mode $2 m + 1$ appears with weight $1 - (2 m + 1) / (N + 1)$ when $2 m + 1 \leq N$ and with weight $0$ otherwise. This is exactly the displayed formula for $σ_{N} q$ .

3.1L1L2step 2.1algebra∎

Every term of the finite sine sum in step 2.1 vanishes at $x = 0$ , so $σ_{N} q (0) = 0$ for every $N$ . Also [L1] and [L2] give $σ_{N} q (x) = \int_{0}^{1} q (x - t) F_{N} (t) d t,$ where the integral is Riemann because $q$ is a step function. Since $∣ q ∣ \leq 1$ , positivity and total mass one imply $∣ σ_{N} q (x) ∣ \leq 1$ for every $x$ . Thus the Fejer means average across the jump and do not exhibit a fixed overshoot.

Depends on

Used by

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Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)
Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)