Pointwise Turán problem for periodic positive definite functions

We study the pointwise Turán problem on the largest value at an arbitrary point $x$ of a $1$-periodic positive definite function supported on the interval $[-h, h]$ and equal to $1$ at zero. For rational values of $x$ and $h$, the problem reduces to a discrete version of the Fejér problem on the largest value of the $\nu$th coefficient of an even trigonometric polynomial of order $p-1$ that has zero coefficient 1 and is nonnegative on a uniform grid $k/q$, $k=0,\dots,q-1$. The discrete Fejér problem is solved for a number of values of the parameters $\nu$, $p$, and $q$. In all the cases, we construct extremal polynomials and quadrature formulas, which yield an estimate for the largest coefficient.