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Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

Authors:

Babenko V. F. Kofanov V. A. Pichugov S. A.

Affiliation:

(1) Dnepropetrovsk University, Dnepropetrovsk

Abstract:

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p isin

[1,

], the following unimprovable inequality holds for functions $$x in L_infty ^r $$

$left| {x^{left( k right)} } right|_q leqslant frac{{left| {{phi }_{r - k} } right|_q }}{{left| {{phi }_r } right|_p^alpha }}left| x right|_p^alpha left| {x^{left( k right)} } right|_infty ^{1 - alpha }$

where $alpha = min left{ {1 - frac{k}{r},frac{{r - k + {1 mathord{left/ {vphantom {1 q}} right. kern-nulldelimiterspace} q}}}{{r + {1 mathord{left/ {vphantom {1 p}} right. kern-nulldelimiterspace} p}}}} right}$ and phiv

_r is the perfect Euler spline of order r.

Keywords:

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