Inequalities of Different Metrics for Differentiable Periodic Functions

  • V. A. Kofanov


We prove the following sharp inequality of different metrics: $$\begin{array}{cc}\hfill {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left(\frac{{\left\Vert x\right\Vert}_p}{{\left\Vert {\varphi}_r\right\Vert}_p}\right)}^{\frac{r+1/q}{r+1/p}}{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{\frac{1/p-1/q}{r+1/p}},\hfill & \hfill q>p>0,\hfill \end{array}$$ for 2π -periodic functions $x ∈ L_{∞}^r$ satisfying the condition $$L{(x)}_p\le {2}^{1/p}{\left\Vert x\right\Vert}_p,$$ where $$L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \left[0,2\pi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\},$$ and $φ_r$ is the Euler spline of order $r$. As a special case, we establish the Nikol’skii-type sharp inequalities for polynomials and polynomial splines satisfying the condition (A).
How to Cite
Kofanov, V. A. “Inequalities of Different Metrics for Differentiable Periodic Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, no. 2, Feb. 2015, pp. 202–212,
Research articles