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Inequalities of Different Metrics for Differentiable Periodic Functions

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Ukrainian Mathematical Journal Aims and scope

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 xL r satisfying the condition

$$ L{(x)}_p\le {2}^{1/p}{\left\Vert x\right\Vert}_p, $$
(A)

where

$$ L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \left[0,2\uppi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\}, $$

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).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 2, pp. 202–212, February, 2015.

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Kofanov, V.A. Inequalities of Different Metrics for Differentiable Periodic Functions. Ukr Math J 67, 230–242 (2015). https://doi.org/10.1007/s11253-015-1076-2

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