Positive definite functions and sharp inequalities for periodic functions

Let $\varphi$ be a positive definite and continuous function on $\mathbb{R}$, and let $\mu$ be the corresponding Bochner measure. For fixed $\varepsilon,\tau\in\mathbb{R}$, $\varepsilon\ne 0$, we consider a linear operator $A_{\varepsilon,\tau}$ generated by the function $\varphi$:
$$A_{\varepsilon,\tau}(f)(t):=\int_{\mathbb{R}}e^{-iu\tau} f(t+\varepsilon u)d\mu(u) ,\quad t\in\mathbb{R},\quad f\in C(\mathbb{T}).$$
Let $J$ be a convex and nondecreasing function on $[0,+\infty)$. In this paper, we prove the inequalities
$$\| A_{\varepsilon,\tau}(f)\|_p\leqslant \varphi(0)\|f\|_p, \quad \int_{\mathbb{T}}J\left(|A_{\varepsilon,\tau}(f)(t)|\right)\,dt \le \int_{\mathbb{T}}J\left(\varphi(0)|f(t)|\right)\,dt$$
for $p\in [1,\infty]$ and $f\in C(\mathbb{T})$ and obtain criteria of extremal function. We study in more detail the case in which $\varepsilon=1/n$, $n\in \mathbb{N}$, $\tau=1$, and $\varphi(x)\equiv e^{i\beta x}\psi(x)$, where $\beta\in\mathbb{R}$ and the function $\psi$ is $2$-periodic and positive definite. In turn, we consider in more detail the case where the 2-periodic function $\psi$ is constructed by means of a finite positive definite function $g$. As a particular case, we obtain the Bernstein–Szegő inequality for the derivative in the Weyl–Nagy sense of trigonometric polynomials. In one of our results, we consider the case of the family of functions $g_{1/n,h}(x):=hg(x)+(1-1/n-h)g(nx)$, where $n\in\mathbb{N}$, $n\ge 2$, $-1/n\le h\le 1-1/n$, and the function $g\in C(\mathbb{R})$ is even, nonnegative, decreasing, and convex on $(0,+\infty)$ with $\mathrm{supp}\ g\subset[-1,1]$. This case is related to the positive definiteness of piecewise linear functions. We also obtain some general interpolation formulas for periodic functions and trigonometric polynomials which include the known interpolation formulas of M. Riesz, of G. Szegő, and of A.I. Kozko for trigonometric polynomials.