On n-K Width of Certain Function ClassesDefined by Linear Operators in L2Space

YU Rui-fang，WU Ga-ridi

（Department of Mathematics，Inner Mongolia Normal University，Huhhote 010022，China）



On n-K Width of Certain Function Classes
Defined by Linear Operators in L2Space

YU Rui-fang，WU Ga-ridi

（Department of Mathematics，Inner Mongolia Normal University，Huhhote 010022，China）

Let M（u）be an N-function，Lr（f，x）and Kr（f，x）are Bak operator and Kantorovich operator，WM（Lr（f））and WM（Kr（f））are the Sobolev-Orlicz classes defined by Lr（f，x），Kr（f，x）and M（u）.In this paper we give the asymptotic estimates of the n-K widths dn（WM（Lr（f）），L2［0，1］）and dn（WM（Kr（f）），L2［0，1］）.

linear operator；Sobolev-Orlicz class；width

2000 MR Subject Classification:41A46

Article ID:1002—0462（2016）01—0096—06

Chin.Quart.J.of Math.

2016，31（1）：96—101

§1.Introduction and Main Result

Let M（u）be an N-function，that is，

（1）M（u）is an even continuous convex function and M（0）=0；

（2）M（u）＞0 for u＞0；

Bak operator Lr（f，x）is defined by

Sobolev-Orlice class WM（Lr（f））is defined by

Kantorovich operator Kr（f，x）is defined by

Sobolev-Orlice class WM（Kr（f））is defined by

WM（Kr（f））=｛f:f（r-1）absolutely continuous on［0，1］，and

For any given natural number r，WM（Lr（f））and WM（Kr（f））are two fixed function classes.

Theorem 1Let M（u）be an N-function satisfying the following conditions，

（1）M（u）is strictly increasing on［0，+∞）；

The cases of linear differential operators

（3）M（u）satisfies∆′-condition，this is，M（uv）≤NM（u）M（v）valid for some N＞0 and all u，v≥0.

Then

Theorem 2Let M（u）be an N-function satisfying the conditions of Theorem 1，then

Where the expression an～bnmeans that C1bn≤an≤C2bnfor some C1，C2＞0.

§2.Auxiliary Lemmas

Lemma 1［4］Assume｛e1，e2，···，ep｝is an orthogonal system of vectors in a Hilbert space H and‖ek‖=h，k=1，2，···，p，then

Lemma 2［2］Assume f（x）（x∈［0，+∞））is a monotonically increasing convex function andwhere x（t）is non-negative continuous on［a，b］，then

§3.Proof of Theorem

Proof of Theorem 1First of all，let us estimate a lower bound of dn（WM（Lr（f）），L2［0， 1］）.We fix an infinitely differentiable function g0（x），such thatfor x∈［0，1］and g0（x）=0 forLet us construct 2n functions

Set

This means that Cfj（x）∈WM（Lr（f）），j=1，2，···，2n，from the Lemma 1，we have

furthermore

Secondly，we estimate an upper bound of dn（WM（Lr（f）），L2［0，1］）.Let us divide interval［0，1］into n equal parts and obtain n subintervals［xj-1，xj），j=0，1，···，n，.For f（x）∈WM（Lr（f）），set

where pr-1（x）is a algebraic polynomial of degree not greater than r-1 and C is any constant. Thenis a linear subspace and dim

For f（x）∈WM（Lr（f）），let us choose a functionthat is

then

where C0is a constant.

Set

By Lemma 2，we obtain

It follows from Lemma 3 and the Condition（2）in Theorem 1 that

By the definition of Kolmogorov width，we obtain

Form this and the monotonicity of M-1（u），we have

Add it all up，we can obtain

The proof of Theorem 1 is finished.

The proof of Theorem 2 is similar to the proof of Theorem 1，we omit it here.

［References］

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O174.41Document code:A

date：2015-06-25

Supported by the National Natural Science Foundation of China（11161033）；Supported by the Inner Mongolia Normal University Talent Project Foundation（RCPY-2-2012-K-036）；Supported by the Inner Mongolia Normal University Graduate Research Innovation Foundation（CXJJS14053）；Supported by the Inner Mongolia Autonomous Region Graduate Research Innovation Foundation（S20141013525）

Biographies：YU Rui-fang（1991-），female（Meng），native of Chifeng，Inner Mongolia，postgraduate，engages in function approximation theory；WU Ga-ridi（1962-），male（Meng），native of Tongliao，Inner Mongolia，Master Instructor，a professors of Inner Mongolia Normal University，engages in function approximation theory.