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
1.1

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.

L1algebra
2.1

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.

step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources