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Fourier partial sums of the sawtooth

Example

Assume the Axiom of Countable Choice.

Let $f$ be the one-periodic sawtooth given by $f (0) = 0$ and $f (x) = x - \frac{1}{2}$ for $0 < x < 1$ . Then $\hat{f} (0) = 0$ and, for $k \neq 0$ ,

$\hat{f} (k) = - \frac{1}{2 π i k} .$

Hence

$S_{N} f (x) = - \sum_{k = 1}^{N} \frac{\sin (2 π k x)}{π k} .$

At every noninteger $x$ , $S_{N} f (x) \to f (x)$ , while at every integer $x$ , $S_{N} f (x) \to 0$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the one-periodic sawtooth $f (0) = 0$ and $f (x) = x - \frac{1}{2}$ for $0 < x < 1$ .

[L1]

Fourier coefficients and partial sums are defined by the one-period formulas in Period-one Fourier coefficients, partial sums, and convolution on the torus.

[L2]

Assuming the Axiom of Countable Choice, a one-periodic bounded-variation function converges at each point to the midpoint of its one-sided limits under Fourier partial sums (Dirichlet-Jordan pointwise convergence).

Verification

technique · direct

1.1

Since $\int_{0}^{1} (x - \frac{1}{2}) d x = 0$ , [L1] gives $\hat{f} (0) = 0$ . For $k \neq 0$ , direct integration gives $\hat{f} (k) = \int_{0}^{1} (x - \frac{1}{2}) e^{- 2 π i k x} d x = - \frac{1}{2 π i k} .$

L1algebra

2.1

Insert the coefficients from step 1.1 into the partial-sum formula [L1]. Pairing the $k$ and $- k$ terms gives $S_{N} f (x) = - \sum_{k = 1}^{N} \frac{\sin (2 π k x)}{π k} .$

L1step 1.1algebra

3.1

The sawtooth is piecewise $C^{1}$ , hence of bounded variation on one period. If $x \notin Z$ , choose the unique integer $m$ with $y : = x - m \in (0, 1)$ . Because both $f$ and its Fourier partial sums are one-periodic, one has $S_{N} f (x) = S_{N} f (y)$ and $f (x) = f (y) = y - \frac{1}{2}$ . The one-sided limits at $y$ therefore both equal $f (x)$ , so [L2] gives $S_{N} f (x) \to f (x)$ . At an integer $x$ , the one-sided limits are $- 1 / 2$ and $1 / 2$ , whose midpoint is $0$ , so [L2] gives $S_{N} f (x) \to 0$ .

L2step 2.1choosealgebra∎

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)