Sharp Remez-Type Inequalities of Various Metrics with Asymmetric Restrictions Imposed on the Functions
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SHARP REMEZ-TYPE INEQUALITIES OF VARIOUS METRICS WITH ASYMMETRIC RESTRICTIONS IMPOSED ON THE FUNCTIONS V. A. Kofanov1 and I. V. Popovich2
UDC 517.5
For any p 2 (0, 1], ! > 0, β 2 (0, 2!), and any measurable set B ⇢ Id := [0, d], µB β, we deduce a sharp Remez-type inequality kx± k1
k(' + c)± k1 kxkLp (Id \B) k' + ckLp (I2! \Byc )
on the set S' (!) of d-periodic functions x with zeros and a given sine-shaped 2!-periodic comparison function '; here, c 2 [−k'k1 , k'k1 ] satisfies the condition −1 kx+ k1 · kx− k−1 1 = k(' + c)+ k1 · k(' + c)− k1 ,
Byc := {t 2 [0, 2!] : |'(t) + c| > y}, and y is such that µByc = β. In particular, inequalities of this type are established on Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient kx+ k1 /kx− k1 .
1. Introduction Let G ⇢ R. We consider spaces Lp (G), 0 < p 1, of all measurable functions x : G ! R with finite norm (quasinorm) kxkLp (G) , where 80 11/p Z > > > p > > for 0 < p < 1,
> > > > :vraisup |x (t)| for p = 1. t2G
Let d > 0, let Id be a circle realized in the form of a segment [0, d] with identified endpoints. For r 2 N and G = R or G = Id , by Lr1 (G) we denote the space of all functions x 2 L1 (G) with locally absolutely continuous derivatives up to the (r − 1)th order such that x(r) 2 L1 (G). For these G, we write kxk1 instead of kxkL1 (G) . By E0 (f )Lp (G) we denote the best approximation of the function f by constants in the space Lp (G), i.e., E0 (x)Lp (G) := inf kx − ckLp (G) . c2R
Further, instead of E0 (x)L1 (G) , we write E0 (x)1 . 1 2
Dnepr National University, Dnepr, Ukraine; e-mail: [email protected]. Dnepr National University, Dnepr, Ukraine.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 918–927, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.2352. Original article submitted February 1, 2020. 1068
0041-5995/20/7207–1068
© 2020
Springer Science+Business Media, LLC
S HARP R EMEZ -T YPE I NEQUALITIES OF VARIOUS M ETRICS WITH A SYMMETRIC R ESTRICTIONS
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We say that f 2 L11 (R) is a comparison function for x 2 L11 (R) if there exists c 2 R such that max x(t) = max f (t) + c, t2R
t2R
min x(t) = min f (t) + c, t2R
t2R
and the equality x(⇠) = f (⌘) + c, where ⇠, ⌘ 2 R, yields the inequality |x0 (⇠)| |f 0 (⌘)| if the indicated derivatives exist. An odd 2!-periodic function ' 2 L11 (I2! ) is called an S-function if it has the following properties: ' is even about !/2 and |'| is convex upward on [0, !] and strictly monotone on [0, !/2]. For a 2!-periodic S-function ', by S' (!) we denote a class of functions x from the space L11 (R) for which ' is a comparison function. Note that the classes S' (!) were considered in [1, 2]. As examples of the classes S' (!), we can mention the Sobolev classes �
x 2 Lr1 (R) : kxk1 A0 , kx(r) k1 Ar
and bounded subsets of the space Tn (of trigonometric polynomials whose degrees do not exceed n) and of the space Sn,r (of periodic splines of order r and defect 1 with nodes at the points k⇡/n, k 2
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