Jackson kernel

Remarks

Since $\left(\frac{\sin(nt/2)}{\sin(t/2)}\right)^2 = n F_{n-1}(t)$, where $F_{n-1}$ is the Fejér kernel, $J_n$ is a constant times $F_{n-1}^2$. So $J_n$ is an even, nonnegative trigonometric polynomial of degree $2n - 2$. Its key property is $\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{t}^k J_n(t) \, dt \le \frac{C}{n^k}$ for $k = 1, 2$, with $C$ independent of $n$.

Example: for $n = 1$, the quotient is $1$, so $J_1 = 1$ and $\lambda_1 = 1$.

Non-example: the Fejér kernel $F_n$ is nonnegative with mean $1$, but it fails the key property for $k = 1$. Its first moment $\frac{1}{2\pi} \int \abs{t} F_n(t) \, dt$ is of order $\frac{\log n}{n}$, not $\frac{1}{n}$.

Texts differ in the indexing; some call $J_m$ with $m \approx n/2$ the kernel of degree $n$. The convolution $f \mapsto f * J_n$ is the Jackson operator.