Summability kernel

Remarks

For an integrable $2\pi$-periodic $f$, the convolutions $k_n * f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} k_n(t) f(x - t) \, dt$ tend to $f$ in $L^1$. If $f$ is continuous, they tend to $f$ uniformly. A kernel with $k_n \ge 0$ is called positive; then clause 2 follows from clause 1 with $C = 1$.

Example: the Fejér kernel $F_N(t) = \sum_{\abs{j} < N} \left(1 - \frac{\abs{j}}{N}\right) e^{ijt} = \frac{1}{N} \left( \frac{\sin(Nt/2)}{\sin(t/2)} \right)^2$. Its mean is the coefficient of $e^{0}$, which is $1$. It is nonnegative, so clause 2 holds with $C = 1$. For $\delta \le \abs{t} \le \pi$ it is at most $\frac{1}{N \sin^2(\delta/2)}$, which gives clause 3.

Non-example: the Dirichlet kernel $D_n(t) = \sum_{\abs{j} \le n} e^{ijt}$ satisfies clause 1, but $\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{D_n}$ grows like a constant times $\log n$, so clause 2 fails.

Kernels indexed by a continuous parameter are treated the same way; the Poisson kernel $P_r$ with $r \to 1^-$ is the main example. A summability kernel is a bounded approximate identity of the convolution algebra $L^1(\mathbb{T})$.