Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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Fejer means need not converge uniformly for discontinuous data

Statement refuted

For every one-periodic integrable function f, the Fejer means σNf converge uniformly to f.

Facts & Assumptions

Given: The one-periodic step function f(x):={1,0<x<1/2,0,−1/2<x<0, with f(0)=f(1/2)=0.

[L1]

The Fejer means are averages of the Fourier partial sums (Cesaro and Abel means of a Fourier series).

Counterexample

technique · direct
1.1L1algebra

For each N, the function Sjf is a trigonometric polynomial for every 0≤j≤N, so [L1] makes σNf a trigonometric polynomial as well. In particular, every σNf is continuous.

2.1step 1.1algebra∎

If σNf converged uniformly to f, then the uniform limit of the continuous functions σNf would be continuous. But the chosen f has a jump at 0, so it is discontinuous. Therefore uniform convergence to f is impossible. This single step function refutes the universal statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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