Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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Uniform Fejer convergence can coexist with divergent ordinary partial sums

Statement refuted

Uniform convergence of the Fejer means of a continuous periodic function forces its ordinary Fourier partial sums to converge at every point.

Assume DC. There is a real continuous one-periodic f such that

∥σNf−f∥∞→0,sup⁡N≥0∣SNf(0)∣=∞.

Facts & Assumptions

Given: DC and the period-one Fourier conventions with normalized Haar measure.

[F1]

Assuming DC, for every prescribed x0∈T there exists a real continuous periodic f with sup⁡N≥0∣SNf(x0)∣=∞ (A continuous function with divergent Fourier series at a prescribed point).

[F2]

For every continuous one-periodic complex function f, sup⁡x∈R∣σNf(x)−f(x)∣→0 (Fejer means converge uniformly for continuous periodic functions).

Counterexample

technique · reuse of the continuous witness and uniform Fejer convergence
1.1

Apply the prescribed-point witness with x0=0. It supplies a real continuous periodic f whose finite partial-sum values at zero form an unbounded sequence. Such an f is nonzero, and this sequence cannot converge to a finite value.

F1given
2.1

Regard the same real f as complex-valued. It meets the continuity and periodicity hypotheses of uniform Fejer convergence, so ∥σNf−f∥∞→0. Thus the Cesaro averages σNf=(N+1)−1∑j=0NSjf converge uniformly while the ordinary sums at zero diverge unboundedly. Both properties hold for the single function from step 1.1, refuting the claimed implication.

F2step 1.1∎

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