Dirichlet and Fejér kernels

Define circle convolution by $(f*g)(x)=\int_{\mathbb T}f(x-t)g(t)\,d\mu(t)$. For integrable $f$, inserting the finite Fourier polynomial $D_N(t)=\sum_{n=-N}^Ne^{int}$ and interchanging a finite sum with the integral gives $S_Nf=f*D_N$. Summing the geometric progression gives a sine quotient away from multiples of the period. The apparently undefined value at zero has the continuous extension $2N+1$, directly visible from the original finite sum.

Dirichlet's kernel integrates to one because its constant coefficient is one, but it takes both signs. Its absolute integral grows on the order of the logarithm of the cutoff, rather than remaining uniformly bounded. Thus the elementary convolution bound based on kernel absolute mass worsens with the number of frequencies. This explains a genuine obstacle to uniform control on all continuous or integrable inputs. It does not say that every particular smooth input is unstable, since smoothness and cancellation can yield much stronger information.

The Fejér mean $\sigma_Nf$ averages $S_0f$ through $S_Nf$, so a frequency with index $n$ appears in exactly $N+1-|n|$ of those sums when $|n|\le N$. Its coefficient weight is consequently $1-|n|/(N+1)$. The corresponding kernel is a squared sine quotient divided by $N+1$. Squaring makes it nonnegative, and its constant coefficient still makes its integral one. Notice the index convention: averaging through $S_N$ produces denominator $N+1$.

For bounded inputs, a positive mass-one kernel bounds the magnitude of the output by the input supremum norm. For integrable inputs it also gives an L¹ contraction, by the integral triangle inequality and translation invariance of circle measure. These statements control differences of inputs because convolution is linear. The price is deliberate attenuation of the highest retained frequencies. A Fejér mean is generally not the orthogonal projection onto its frequency range, so improved averaging stability should not be confused with the best fixed-degree L² approximation property.

For every fixed neighborhood of zero on the circle, the Fejér kernel's integral outside that neighborhood tends to zero. The denominator in its explicit formula stays away from zero there, while the numerator is bounded and the factor in front shrinks. Positivity, unit mass, and this concentration show uniform convergence to every continuous periodic input: split the convolution error into a small neighborhood controlled by uniform continuity and a far region controlled by its shrinking mass.

This proof describes an approximate identity. It does not mean the kernels converge to an ordinary integrable function concentrated at one point; instead they approach the identity operation in specified function spaces, or a point mass in a weak measure sense. For Lᵖ inputs with 1≤p<∞, density and uniform convolution bounds provide norm convergence. Arbitrary bounded measurable functions need not have uniform convergence. A jump remains a place where the chosen averaging rule, the nearby limits, and the assigned point value must be distinguished.