Abstract
For any $p\in (0, \infty],$ $\omega > 0,$ $d \ge 2 \omega,$ we obtain the sharp inequality of Nagy type$$\|x_{\pm}\|_\infty \le\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega})}} \left\|x \right\|_{L_{p} \left(I_d \right)}$$on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodiccomparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ is such that$$ \|x_{+}\|_\infty \cdot\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot\|(\varphi+c)_{-}\|^{-1}_\infty .$$In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $\|x_{+}\|_\infty / \|x_-\|_\infty$.
Highlights
Let d > 0 and Id denote the circle which is realized as the interval [0, d] with coincident endpoints
Let us recall that Tn is the space of all trigonometric polynomials of degree at most n
By conditions of Theorem 4 for a spline s ∈ Sn,r with minimal period
Summary
We will consider the spaces Lp(G), 0 < p ≤ ∞, of all measurable functions x : G → R such that x p = x Lp(G) < ∞, where x p :=. For r ∈ N, G = R or G = Id, denote by Lr∞(G) the space of all functions x ∈ L∞(G) for which x(r−1) is locally absolutely continuous and x(r) ∈ L∞(G). For 2ω-periodic S-function φ denote by Sφ(ω) the class of functions x ∈ L1∞(R) for which φ is the comparison function. Note that the classes Sφ(ω) were considered in [1], [2]. Examples of such classes Sφ(ω) are the Sobolev classes {x ∈ Lr∞(Id) : x(r) ∞ ≤ 1}, the bounded subsets of the space Tn of all trigonometric polynomials of order at most n, and the same subsets of the space Sn,r of polynomial splines of order r having defect 1 with knots at the points kπ/n, k ∈ Z
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