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The Fejer kernel is a positive approximate identity

Statement

For every $N \geq 0$ , the Fejer kernel satisfies

$F_{N} (t) = \frac{1}{N + 1} {∣ \sum_{j = 0}^{N} e_{j} (t) ∣}^{2} .$

Hence, for $t \notin Z$ ,

$F_{N} (t) = \frac{1}{N + 1} {(\frac{\sin ((N + 1) π t)}{\sin (π t)})}^{2},$

so $F_{N} (t) \geq 0$ for all $t$ , $\int_{0}^{1} F_{N} (t) d t = 1$ , and for every $δ \in (0, 1 / 2]$ ,

$\int_{δ}^{1 - δ} F_{N} (t) d t \leq \frac{1}{(N + 1) \sin^{2} (π δ)} .$

In particular,

$\int_{δ}^{1 - δ} F_{N} (t) d t ⟶ 0 (N \to \infty) .$

Facts & Assumptions

Given: An integer $N \geq 0$ and a real $δ \in (0, 1 / 2]$ .

[L1]

The Fejer kernel is $F_{N} (t) = \frac{1}{N + 1} \sum_{m = 0}^{N} D_{m} (t)$ , where $D_{m} (t) = \sum_{∣ k ∣ \leq m} e_{k} (t)$ and $\int_{0}^{1} D_{m} (t) d t = 1$ (Dirichlet and Fejer kernels).

Proof

technique · direct

1.1

Expanding the average in [L1] gives $\sum_{m = 0}^{N} D_{m} (t) = \sum_{m = 0}^{N} \sum_{∣ k ∣ \leq m} e_{k} (t) = \sum_{∣ k ∣ \leq N} (N + 1 - ∣ k ∣) e_{k} (t) .$ On the other hand, ${∣ \sum_{j = 0}^{N} e_{j} (t) ∣}^{2} = \sum_{j = 0}^{N} \sum_{ℓ = 0}^{N} e_{j - ℓ} (t) = \sum_{∣ k ∣ \leq N} (N + 1 - ∣ k ∣) e_{k} (t) .$ Therefore $F_{N} (t) = \frac{1}{N + 1} {∣ \sum_{j = 0}^{N} e_{j} (t) ∣}^{2} .$

L1algebra

2.1

If $t \notin Z$ , the finite geometric-series formula gives $\sum_{j = 0}^{N} e_{j} (t) = \sum_{j = 0}^{N} e^{2 π i j t} = e^{π i N t} \frac{\sin ((N + 1) π t)}{\sin (π t)},$ so step 1.1 yields the displayed square formula. This proves $F_{N} (t) \geq 0$ for every $t \notin Z$ , and at integers the same formula extends by continuity to $F_{N} (0) = N + 1$ . Also [L1] gives $\int_{0}^{1} F_{N} (t) d t = \frac{1}{N + 1} \sum_{m = 0}^{N} \int_{0}^{1} D_{m} (t) d t = 1.$

L1step 1.1algebra

3.1

For $t \in [δ, 1 - δ]$ , one has $∣ \sin (π t) ∣ \geq \sin (π δ)$ . Using step 2.1 and $∣ \sin ((N + 1) π t) ∣ \leq 1$ therefore gives $F_{N} (t) \leq \frac{1}{(N + 1) \sin^{2} (π δ)} .$ Integrating over an interval of length at most $1$ yields $\int_{δ}^{1 - δ} F_{N} (t) d t \leq \frac{1}{(N + 1) \sin^{2} (π δ)} \to 0.$

step 2.1algebra∎

Depends on

Dirichlet and Fejer kernels

Used by

Fejer summation of the square wave Example
Gibbs phenomenon Remark
Fejer means converge at Lebesgue points Theorem
Fejer means converge in Lᵖ for 1 <= p < infinity Theorem
Fejer means converge to midpoint values at jumps Theorem
Fejer means converge uniformly for continuous periodic functions Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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