= Fejér's theorem
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{title2=$\sigma_Nf\to f$}
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= Fejér theorem
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The <Cesàro means> of the <Fourier series> of a continuous periodic function converge uniformly to that function. The <Fejér kernel> is nonnegative with total mass one after normalization, and its mass outside every fixed neighborhood of zero tends to zero. Uniform <continuity> controls the near contribution, while the vanishing tail controls the rest. For an $L^1$ function the same positive-kernel argument gives an $L^1$ contraction, which is useful even when pointwise behavior is unavailable.
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