Sharp Remez-Type Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

We prove a sharp Remez-type inequality of different metrics

$$ {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left\{\frac{{\left\Vert x\right\Vert}_{L_p\left(\left[0,2\uppi \right]\backslash B\right)}}{{\left\Vert {\varphi}_r\right\Vert}_{L_p\left(\left[0,2\uppi \right]\backslash {B}_1\right)}}\right\}}^{\alpha }{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{1-\alpha },\kern0.5em q>p>0,\kern0.5em \alpha =\left(r+1/q\right)/\left(r+1/p\right), $$

for 2π -periodic functions x ∈ ${L}_{\infty}^r$ satisfying the condition

where $L{(x)}_p\coloneq \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:\left[a,b\right]\subset \left[0,2\uppi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\}$, B ⊂ [0, 2π], μB ≤ β/λ (λ is chosen so that ${\left\Vert x\right\Vert}_p={\left\Vert {\varphi}_{\uplambda, r}\right\Vert}_{L_p\left[0,2\uppi /\uplambda \right]}$), φ _r is the ideal Euler’s spline of order r, and ${B}_1\coloneq \left[\frac{-\uppi -\beta /2}{2},\frac{-\uppi +\beta /2}{2}\right]\cup \left[\frac{\uppi -\beta /2}{2},\frac{\uppi +\beta /2}{2}\right]$.

As a special case, we establish sharp Remez-type inequalities of different metrics for trigonometric polynomials and polynomial splines satisfying the condition (_*).