SWhat changed
Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means.
Scientific scope
Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means. The averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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