Fejér's theorem

For every continuous periodic function, the arithmetic means of its Fourier partial sums converge uniformly to the function. These means are called its Fejér means.

Fejér kernel: A nonnegative trigonometric polynomial whose convolution with a periodic function gives its Fejér mean. Its positivity and concentration make the averaged sums converge uniformly.
Fourier series: A representation of a periodic function as a sum of sine and cosine waves, or complex exponentials. The theorem concerns convergence of averaged partial sums of this expansion.
László Fejér: A Hungarian mathematician known for foundational work on Fourier series and summability. He proved the uniform convergence result for Cesàro means of continuous periodic functions.
Weierstrass approximation theorem: Every continuous function on a closed bounded interval can be uniformly approximated by polynomials. Fejér means give a trigonometric route to uniform approximation for continuous periodic functions.
Uniform approximation: Approximation in which the maximum error over a whole domain can be made arbitrarily small. Fejér means establish this strong form of approximation for continuous periodic functions.
Cesàro summation: A summation method that replaces a sequence's terms by averages of its partial sums. Fejér means apply first-order Cesàro averaging to Fourier partial sums.
Fourier coefficient: A scalar measuring a periodic function's component at a particular frequency. The coefficients determine the partial sums that Fejér averaging combines.
Fourier convergence: The study of whether and how Fourier expansions approach the functions they represent. Fejér's theorem gives a robust convergence result where ordinary partial sums can fail.
Gibbs phenomenon: Persistent overshoot near a jump discontinuity in Fourier partial sums. Averaging suppresses oscillatory overshoot, though the theorem assumes continuity.
Fourier series divergence: Failure of Fourier partial sums to converge at some points or in a specified sense. Fejér's theorem shows averaging can converge uniformly even when raw partial sums behave poorly.

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Dirichlet–Jordan testCompared with: Averaging partial sums can ensure convergence under conditions different from bounded variation.