Probabilistic linear widths of Sobolev space with Jacobi weights on \([ 1,1]\)

R E S E A R C H

Open Access

Probabilistic linear widths of Sobolev

space with Jacobi weights on

[–, ]

Xuebo Zhai

*

_{and Xiuyan Hu}

*_{Correspondence:}

[email protected] School of Mathematical and Statistics, Zaozhuang University, Zaozhuang, 277160, China

Abstract

Optimal asymptotic orders of the probabilistic linear (n,

δ

)-widths of

λ

n,δ(W_2,r_α,β,

ν

,Lq,α,β) of the weighted Sobolev spaceW_2,r_α,βequipped with a Gaussian measure

ν

are established, whereLq,α,β, 1≤q≤ ∞, denotes theLqspace on [–1, 1]

with respect to the measure (1 –x)α(1 +x)β,

α

,

β

> –1/2.

MSC: 41A46; 41A25; 28C20; 42C15

Keywords: probabilistic linear widths; Jacobi weights; weighted Sobolev classes; Gaussian measure

1 Introduction

This paper mainly focuses on the study of probabilistic linear (n,δ)-widths of a Sobolev space with Jacobi weights on the interval [–, ]. This problem has been investigated only recently. For calculation of probabilistic linear (n,δ)-widths of the Sobolev spaces equipped with Gaussian measure, we refer to [–]. Let us recall some definitions.

LetKbe a bounded subset of a normed linear spaceXwith the norm · X. The linear

n-width of the setKinXis defined by

λn(K,X) =inf

Lnsup_x∈Kx–LnxX,

whereLnruns over all linear operators fromXtoXwith rank at mostn.

LetW be equipped with a Borel fieldBwhich is the smallestσ-algebra containing all open subsets. Assume thatν is a probability measure defined onB. Letδ∈[, ). The probabilistic linear (n,δ)-width is defined by

λn,δ(W,ν,X) =inf

Gδλn(W\Gδ,X),

whereGδruns through all possibleν-measurable subsets ofW with measureν(Gδ)≤δ.

Compared with the classical case analysis (see [] or []), the probabilistic case analysis, which reflects the intrinsic structure of the class, can be understood as theν-distribution of the approximation on all subsets ofWbyn-dimensional subspaces and linear operators with rankn.

weightedLqspace. Motivated by Wang’s work, this paper considers the probabilistic

lin-ear (n,δ)-widths on the interval [–, ] with Jacobi weights and determines the asymptotic orders of the probabilistic linear (n,δ)-widths. The difference between the work of Wang and ours lies in the different choices of the weighted points for the proofs of discretization theorems.

2 Main results

Consider the Jacobi weights

wα,β(x) := ( –x)α( +x)β, α,β> –/.

Denote by Lp,α,β≡Lp(wα,β), ≤p<∞, the space of measurable functions defined on

[–, ] with the finite norm

fp,α,β:=



–

f(x)pwα,β(x)dx

/p

, ≤p<∞,

and forp=∞we assume thatL∞,α,βis replaced by the spaceC[–, ] of continuous

func-tions on [–, ] with the uniform norm. Letnbe the space of all polynomials of degree

at mostn. Denote byPnthe space of all polynomials of degreenwhich are orthogonal to polynomials of low degree inL(wα,β). It is well known that the classical Jacobi

polynomi-als{P(nα,β)}∞n=form an orthogonal basis forL,α,β:=L([–, ],wα,β) and are normalized by

P(nα,β)() =

n+α n

(see []). In particular,



–

P(_nα,β)(x)P(_nα,β)(y)wα,β(x)dx=δn,mhn(α,β),

where

hn(α,β) =

(α+β+ ) (α+ ) (β+ )

(n+α+ ) (n+β+ )

(n+α+β+ ) (n+ ) (n+α+β+ )∼n

–

with constants of equivalence depending only onα andβ. Then the normalized Jacobi polynomialsPn(x), defined by

Pn(x) =

h(α,β)

n

–/

P(α,β)

n (x), n= , , . . . ,

form an orthonormal basis forL,α,β, where the inner product is defined by

f,g := 

–

f(x)g(x)wα,β(x)dx.

Denote bySnthe orthogonal projector ofL(wα,β) ontoninL(wα,β), which is called the

Fourier partial summation operator. Consequently, for anyf ∈L(Wα,β),

f = ∞

l=

f,Pl Pl, Snf:= n

l=

It is well known that (see Proposition .. in [])Pn(α,β)is just the eigenfunction

corre-sponding to the eigenvalues –n(n+α+β+ ) of the second-order differential operator

Dα,β:=

 –xD–α–β+ (α+β+ )xD, which means that

Dα,βP(nα,β)(x) = –n(n+α+β+ )P( α,β)

n (x).

Givenr> , we define the fractional power (–Dα,β)r/of the operator –Dα,βonf by

(–Dα,β)r/(f) =

∞

k=

k(k+α+β+ )r/f,Pk Pk,

in the sense of distribution. We callf(r)_{:= (–D}

α,β)r/therth order derivative of the

distri-butionf. It then follows that forf∈L,α,β,r∈R, the Fourier series of the distributionf(r)

is

f(r)= ∞

k=

k(k+α+β+ )r/f,Pk Pk.

Using this operator, we define the weighted Sobolev class as follows: Forr>  and ≤ p≤ ∞,

W_pr_,α,β

[–, ]≡W_pr_,α,β:= f ∈Lp,α,β:fW_pr_,α,β:=fp,α,β+(–Dα,β) r

_(f)

p,α,β<∞

,

while the weighted Sobolev classBW_pr_,α,β is defined to be the unit ball ofWpr,α,β. When

p= , the norm · Wr

,α,β is equivalent to the norm · Wr,α,β, and we can rewriteW r

,α,β

as

W_,rα,β =W r

,α,β

:=

f(x) = ∞

l=

f,Pn Pn(x) :f_Wr

,α,β:=f,P

_+f(r)_,f(r)

=f,P +

∞

k=

k(k+α+β+ )rf,Pk <∞

with the inner product

f,g r:=f,P g,P +

f(r),g(r).

Obviously,Wr_,α,β is a Hilbert space. We equipW r

,α,β=W,rα,βwith a Gaussian measure

ν whose mean is zero and whose correlation operatorCν has eigenfunctionsPl(x), l=

, , , . . . , and eigenvalues

λ= , λl=

that is,

CνP=P, CνPl=λlPl, l= , , . . . .

Then (see [], pp.-),

Cνf,g r=

Wr,α,β

f,hrg,hrν(dh).

By Theorem .. of [] the Cameron-Martin spaceH(ν) of the Gaussian measureνis Wr_,+_αs/_,β, i.e.,

H(ν) =Wr_,+_αs/_,β.

See [] and [] for more information about the Gaussian measure on Banach spaces. Throughout the paper,A(n,δ)B(n,δ) meansA(n,δ)B(n,δ) andA(n,δ)B(n,δ), A(n,δ)B(n,δ) means that there exists a positive constantcindependent ofnandδsuch that A(n,δ)≤cB(n,δ). If ≤q≤ ∞,r> ( + min{,max{α,β}})(/p– /q)+, the space

Wr

p,α,βcan be continuously embedded into the spaceLq,α,β(see Lemma . in []).

Setρ=r+_s. The main result of this paper can be formulated as follows.

Theorem . Let≤q≤ ∞,δ∈(, /],and letρ> / + (max{α,β}+ )(/ + /q)+.

Then

λn,δ

W_,r_α,β,ν,Lq,α,β

⎧ ⎨ ⎩

n/–ρ_{( +n}–min{/,/q}_)(ln(

δ))



_{, }≤_q<∞_,

n/–ρ_(ln(n δ))



, q=∞.

(.)

For the proof of Theorem ., the discretization technique is used (see [, , , ]). Since the known results of the probabilistic linear widths of the identity matrix onRm_are

inappropriate here, the probabilistic linear widths of diagonal matrixes onRm_{are adopted}

for the proof of the upper estimates.

3 Main lemmas

Letm

q (≤q≤ ∞) denote the spaceRmequipped with themq-norm defined by

xmq :=

⎧ ⎨ ⎩

(m_i=|xi|q)



q_{, }≤_q<∞_,

max≤i≤m|xi|, q=∞.

We identifyRm_{with the space}m

, denote byx,y the Euclidean inner product ofx,y∈Rm,

and write · instead of · m_.

Consider inRm_{the standard Gaussian measureγ}

m, which is given by

γm(G) = (π)–m/

G exp–x

whereGis any Borel subset inRm_{. Let ≤q≤ ∞, ≤n<m, andδ∈[, ). The}

proba-bilistic linear (n,δ)-width of a linear mappingT:Rm_→lm

q is defined by

λn,δ

T:Rm→lm_q,γm

=inf

GδinfTn_Rsupm_\GδTx–Tnxl m q,

whereGδruns over all possible Borel subsets ofRmwith measureγm(Gδ)≤δ, andTnruns

over all linear operators fromRm_tolm

q with rank at mostn.

Throughout the paper,Ddenotes them×mreal diagonal matrixdiag(d, . . . ,dm) with

d≥d≥ · · · ≥dm> ,Dndenotes them×mreal diagonal matrixdiag(d, . . . ,dn, , . . . , )

with ≤n≤m, andImdenotes them×midentity matrix. Moreover,{e, . . . ,em}denotes

the standard orthonormal basis inRm:

e= (, , . . . , ), . . . , em= (, . . . , , ).

Now, we introduce several lemmas which will be used in the proof of Theorem ..

Lemma .

() (See[])If≤q≤,m≥n,δ∈(, /],then

λn,δ

Im:Rm→lmq,γm

m/q+m/q–/ln(/δ). (.)

() (See[])If≤q<∞,m≥n,δ∈(, /],then

λn,δ

Im:Rm→lmq,γm

m/q+ln(/δ). (.)

() (See[])Ifq=∞,m≥n,δ∈(, /],then

λn,δ

Im:Rm→lmq,γm

ln(m–n)/δlnm+ln(/δ). (.)

Lemma .(See []) Assume that

m

i=

dβ_i ≤C(m,β) for someβ> .

Then,for≤q≤ ∞,m≥n,δ∈(, /],we have

λn,δ

D:Rm→lm_q,γm

C(m,β)

n+  

β

⎧ ⎨ ⎩

(m/q₊√_{ln(/δ), ≤q<∞,} √

lnm+ln(/δ), q=∞. (.)

Letξj=cosθj, ≤j≤n, denote the zeros of the Jacobi polynomialP( α,β)

n (t), ordered so

that

 =:θ<θ<· · ·<θn<θn+:=π.

Letλn(t) be the Christoffel function andbj=λn(ξj). Denote

W(n;ξj) =

 –x+n–α+



_{ –x+n}–β+_

It is well known uniformly (see [])

θj+–θjn–, θjjn– (≤j≤n),

and also

bjn–wα,β(ξj)

 –ξ_j/n–W(n;ξj),

where the constants of equivalence depend only onα,β(see [] or []). The following lemma is well known as Gaussian quadrature formulae.

Lemma .(See []) For each n≥,the quadrature



–

f(x)wα,β(x)dx

n

j=

bjf(ξj) (.)

is exact for all polynomials of degreen– .Moreover,for any≤p≤ ∞,f∈n,we have

fp,α,β

_n

j=

bjf(ξj) p

/p

. (.)

An equivalence like (.) is generally called a Marcinkiewicz-Zygmund type inequality.

Lemma .(See [], Lemma .) Letα,β> –/,σ∈(, 

max{α,β}+)and let bj, ≤j≤n,

be defined as in Lemma..Then

n

j=

b–_jσn+σ. (.)

Let

Ln(x,y) :=

∞

j=

η

j n

Pj(x)Pj(y), x,y∈[–, ], (.)

where η∈C∞(R) is a nonnegativeC∞-function on [,∞) supported in [, ] with the properties thatη(t) =  for ≤t≤ andη(t) >  fort∈[, ). For anyf ∈L,α,β, we define

δ(f) =S(f), δk(f) =Sk(f) –S_k–(f) fork= ,  . . . , (.)

whereSnis given in (.). Denote by

Mk(x,y) =

k

l=k–_+

Pl(x)Pl(y) (.)

the reproducing kernel of the Hilbert spaceL,α,β∩

k

n=k–_+Pn. Then, forx∈[, ],

δk(f)(x) =

k

l=k–_+



–

f(x)Pl(x)Pl(y)wα,β(y)dx=

f,Mk(·,x)

Forf∈_nk_=k–_+Pn,

f(x) =δk(f)(x) =

f,Mk(·,x)

.

By Lemma ., there exists a sequence of positive numberswi=bin–Wα,β(n;ξi), ≤

i≤k+, for which the following quadrature formula holds for allf∈_k+_–:



–

f(t)Wα,β(t)dt=

k+

i=

wif(ξi). (.)

Moreover, for any ≤p≤ ∞,f∈_k, we have

fp,α,β

_k+

i=

wif(ξi) p

/p

=Un(f)k+

p,w ,

wherew= (w, . . . ,wk+),Uk:k−→R k+

is defined by

Uk(f) =

f(ξ), . . . ,f(ξk+)

, (.)

and forx∈Rk+_,

xk+

p,w :=

⎧ ⎨ ⎩

(_i=k+|xi|pwi)



p_{, }≤_p<∞_,

max_{≤i≤}k+|xi|, p=∞.

Let the operatorTk:R k+

−→_k+be defined by

Tka(x) :=

k+

i=

aiwiLk+(x,ξ_i), (.)

wherea:= (a, . . . ,ak+)∈R

k+

. It is shown in [] that for ≤q≤ ∞,

Tkaq,α,β vk+

q,w . (.)

Forf∈_k+, we have

f(x) = 

–

f(y)L_k+(x,y)wα,β(x,y)dy=

k+

i=

wif(ξi)Lk+(x,ξ_i) =T_kU_k(f)(x).

In what follows, we use the lettersSk,Rk,Vkto denoteuk×ukreal diagonal matrixes as

follows:

Sk=diag

w

 

, . . . ,w

 

k+

,

Rk=diag

w



q

, . . . ,w



q

k+

,

Vk=diag

w–

 +q

 , . . . ,w –_+q

k+

,

(.)

Lemma . For any z= (z, . . . ,zk+)∈R

k+

,we have

k+

j=

w

 

j zjMk(·,ξj)

,α,β

z

l_k+, (.)

where Mk(x,y)is given in(.),and(ξ, . . . ,ξk+)is defined as above.

Proof Denote byKthe set

g∈

k

j=k–_–

Pj:g,α,β≤

.

Since

k+

j=

w/_j zjMk(·,ξj)∈L,α,β∩

_k

j=k–_–

Pj

.

By the Riesz representation theorem and the Cauchy-Schwarz inequality, we have

k+

j=

w/_j zjMk(·,ξj)

,α,β

= sup g∈K

_k+

j=

w/_j zjMk(·,ξj),g

= sup g∈K

k+

j=

w/_j zjg(ξj)

≤ sup g∈K

_k+

j=

|zj|

/_k+

j=

wjg(ξj)

/

sup g∈K

_k+

j=

|zj|

/

g,α,β

≤ z

l_k+.

4 Proofs of main results

Before Theorem . is proved, we establish the discretization theorems which give the reduction of the calculation of the probabilistic widths.

Theorem . Let≤q≤ ∞,σ∈(, ),and let the sequences of numbers{nk}and{σk}be

such that≤nk≤k+=:mk,

∞

k=nk≤n,σk∈(, ),

∞

k=σk≤σ.Then

λn,σ

W_,rα,β,ν,Lq,α,β

≤

∞

k=

–kρλnk,σk

Vk:Rmk→lmkq ,γmk

. (.)

Proof For convenience, we write

λnk,σk:=λnk,σk

Vk:Rmk→lmkq ,γmk

whereγmk is the standard Gaussian measure inR

mk_{. Denote byL}

ka linear operator from Rmk _toRmk _{such that the rank ofL}

kis at mostnkand

γmk y∈Rmk|Vky–Lky> λnk,σk

≤σk.

Then, for anyf∈W_,rα,β, by (.)-(.), (.) and (.) we have

δk(f) –TkR–k LkSkUkδk(f)_q,α,β =TkUkδk(f) –TkR–k LkSkUkδk(f)_q,α,β ≤Ukδk(f) –R–k LkSkUkδk(f)_lmk

q,w

=VkSkUkδk(f) –LkSkUkδk(f)_lmk

q . (.)

Lety=SkUkδk(f) = (w

 

δk(f)(ξ), . . . ,w

 

mkδk(f)(ξmk))∈Rmk, forx∈[–, –],

δk(f)(x) =

f,Mk(·,x)

=f(–r),M_k(–r,)(·,x)_r=f,M_k(–r,)(·,x)_r,

whereM(r,)

k (x,y) is ther-order partial derivative ofMk(x,y) with respect to the variable

x,r∈R. Since the random vectorf inW,rα,βis a centered Gaussian random vector with

a covariance operatorCν, the vector

y=SkUkδk(f) =

f,w

 

M (–r,)

k (·,ξ)

r, . . . ,w

 

mkMk(–r,)(·,ξmk)r

inRmk_{is a random vector with a centered Gaussian distributionγ inR}mk_{, and its}

covari-ance matrixCγ is given by

Cγ =

Cν w   i M

(–r,)

k (·,ξi)

,w

 

j M

(–r,)

k (·,ξj)

r

mk i,j=.

Since for anyz= (z, . . . ,zmk)∈R mk_, mk j= w  

j zjMk(·,ξj)∈

k

j=k–_+

Pj, and Cν w   i M

(–r,)

k (·,ξi)

,w

 

j M

(–r,)

k (·,ξj)

r= w   i M

(–r–s,)

k (·,ξi),w

 

jM

(–r,)

k (·,ξj)

r

=w

 

i M

(–ρ,)

k (·,ξi),w

 

j M

(–ρ,)

k (·,ξj)

,

by Lemma . we get

Rmk(y,z)

_{γ(dy) = zC}

γzT= mk

i,j=

zizj

w

 

iM

(–ρ,)

k (·,ξi),w

 

j M

(–ρ,)

k (·,ξj)

= _m k j= w  

j zjMk(–ρ,)(·,ξj), mk j= w  

j zjMk(–ρ,)(·,ξj)

=

mk

j=

w

 

j zjM(–_kρ,)(·,ξj)





–kρ

mk

j=

w

 

j zjMk(·,ξj)





–kρz_lmk

 = 

–kρ

Rmk(y,z)

_γ

mk(dy). (.)

Now we consider the subset ofW_,rα,β

Gk:= f ∈W,rα,β|δk(f) –TkR–k LkSkUkδk(f)_lmk_q > cc–kρλnk,σk

,

wherec,care the positive constants given in (.), (.). Then by (.) we get

ν(Gk)≤ν f ∈W,rα,β|VkSkUkδk(f) –LkSkUkδk(f)_lmk q > c

–kρ_λ nk,σk

=γ y∈Rmk|Vky–Lky_lmk q > c

–kρ_λ nk,σk

.

Note that for anyt> , the set{y∈Rmk_:V

ky–Lky_lmk

q ≤t}is convex symmetric. It then

follows by Theorem .. in [] and (.), we have

ν(Gk)≤γ y∈Rmk:Vky–Lky_lmk_q > c–kρλnk,σk

≤λ y∈Rmk:Vky–Lky_lmk_q > c–kρλnk,σk

≤γmk y∈R mk_:V

ky–Lky_lmk

q > λnk,σk

≤σk,

whereλis a centered Gaussian measure inRmk _{with covariance matrixc}

–kρImk.

Con-siderG=∞_k=Gkand the linear operatorTnonW,rα,βwhich is given by

Tnf =

∞

k=

TkR–k LkSkUkδk(f).

Then

ν(G) =ν

_∞

k=

Gk

≤

∞

k=

ν(Gk)≤

∞

k=

ν(σk)≤σ,

and

rankTn≤

∞

k=

rankTkR–k LkSkUkδk

≤

∞

k=

nk≤n.

Thus, according to the definitions ofG,Tn, andLk, we obtain

λn,δ

W_,rα,β,ν,Lq,α,β

= sup f∈W_,r_α,β\G

f –Tnfq,α,β

≤ sup f∈W_,r_α,β\G

∞

k=

≤

∞

k=

sup f∈W_,r_α,β\G

δk(f) –TkR–k LkSkUkδk(f)_q,α,β

∞

k=

–kρλnk,σk,

which completes the proof of Theorem ..

Now we turn to the lower estimates. Assume thatm≥ andbm≤n≤bmwithb> 

being independent ofnandm. Set{xj}Nj=⊂ {x∈[–, ] :|x| ≤/}andxj+–xj= /m,

j= , . . . ,N– . ThenMNand

x∈[–, ] :|x–xj| ≤/m

∩ x∈[–, ] :|x–xi| ≤/m

=∅, ifi=j.

We may takeb>  sufficiently large so thatN≥n. Letϕ be aC∞-function onR

sup-ported in [–, ], and be equal to  on [–/, /]. Letϕ_{be a nonnegativeC}∞_{-function on}

Rsupported in [–/, /], and be equal to  on [–/, /]. Define

ϕi(x) =ϕ

m(x–xi)

–ciϕ

m(x–xi)

,

for somecisuch that



–ϕi(x)Wα,β(x)dx= ,i= , . . . ,N. Set

AN:=span{ϕ, . . . ,ϕN}=

Fa(x) =

N

j=

ajϕj(x) :a= (a, . . . ,aN)∈RN

.

Clearly,

ϕj∈W,α,β, suppϕj⊂ x∈[–, ] :|x–xj| ≤/m

⊂ x∈[–, ] :|x| ≤/,

ϕjq,α,β

/

–/

ϕj(x) q

dx /q

= /

–/

ϕm(x–xj)

–cjϕ

m(x–xj) q

dx /q

m–/q, ≤q≤ ∞,j= , . . . ,N,

and

suppϕj∩suppϕi=∅ (i=j).

It follows that forFa∈An,a= (a, . . . ,aN)∈RN, we have

Faq,α,β

m–

N

j=

|aj|q

/q

=m–/qa_lN

q. (.)

For a nonnegative integerν= , , . . . , andFa∈AN,a= (a, . . . ,aN)∈RN, it follows from

the definition of –Dα,βthat

supp(–Dα,β)ν(ϕj)⊂ x∈[–, ] :|x–xj| ≤/m

and

(Dα,β)ν(ϕj)_q,α,β≤m

ν–/q_.

Hence, for ≤q≤ ∞andFa=

N

j=ajϕj∈AN,

(–(Dα,β)ν(Fa)_q,α,β≤mν–/qalN q.

It then follows by the Kolmogorov type inequality (see Theorem . in []) that

F_a(ρ)_q,α,β = (–Dα,β)ρ/(Fa)_q,α,β

(–Dα,β)+[ρ](Fa) ρ

+[ρ]

q,α,β Fa

–_+[ρ_ρ]

q,α,β mρ–/qa_lN

q m

ρ

Faq,α,β. (.)

Forf∈L,α,βandx∈[–, ], we define

PN(f)(x) = N

j=

ϕj(x) ϕj,α,β



–

f(y)ϕj(y)Wα,β(y)dy

and

QN(f)(x) = N

j=

ϕj(x) ϕj,α,β



–

f(y)ϕ_j(ρ)(y)Wα,β(y)dy.

Clearly, the operatorPN is the orthogonal projector fromL,α,β toAN, and iff ∈W,ρα,β,

thenQN(f)(x) =PN(fρ)(x). Also, using the method in [], we can prove thatPN is the

bounded operator fromLq,α,βtoAN∩Lq,μfor ≤q≤ ∞,

_PN(f)

q,α,β fq,α,β. (.)

SinceQN(f)∈AN forf ∈W,ρα,β, we have

_QN(f)(ρ)

,α,βm ρ

QN(f))_,α,β=mρPN(f)(ρ)_,α,βmρf(ρ)_,α,β. (.)

Theorem . Let≤q≤ ∞,δ∈(, ),and let N be given above.Then

λn,δ

W_,rα,β,ν,Lq,α,β

n/–ρ–/qλn,δ

IN:RN→lqN,γN

,

where Nn,N≥n andγN is the standard Gaussian measure inRN.

Proof LetTnbe a bounded linear operator onW,rα,βwith rankTn≤nsuch that

ν f ∈W_,rα,β:f–Tnfq,α,β> λn,δ

whereλn,δ:=λn,δ(W_,rα,β,ν,Lq,α,β). Note that ifAis a bounded linear operator fromW_,rα,β

toW_,rα,βand fromH(ν) toH(ν), then the image measureλofνunderAis also a centered

Gaussian measure onW_,rα,βwith covariance

Rλ(f)(f) =

A∗Cνf,A∗Cνf

H(ν), f ∈W

r

,α,β,

whereCνis the covariance of the measureν,H(ν) =W ρ

,α,β is the Camera-Martin space

ofν, andA∗ is the adjoint ofAinH(ν) (see Theorem .. of []). Furthermore, if the operatorAalso satisfies

AfH(ν)≤ fH(ν),

then

Rλ(f)(f) =A∗Cνf



H(ν)≤A

∗

Cνf ≤ Cνf,Cνf H(ν)=Rν(f)(f).

By Theorem .. in [], we get that for any absolutely convex Borel setEofW_,rα,βthere

holds the inequality

ν(E)≤λ(E).

Applying (.) we assert that

QN(f)_H(ν)=QN(f)

(ρ)

,α,βm ρ_f(ρ)

,α,β=m ρ_f

H(ν).

Then there exists a positive constantcsuch that

_c_mρQN(f)

H(ν)

≤ fH(ν).

Note that, for anyt> , the set{f∈W_,rα,β:f–Tnfq,α,β≤t}is absolutely convex. It then

follows that

ν f ∈W_,rα,β:f–Tnfq,α,β< λn,δ

≤λ f ∈W_,rα,β:f –Tnfq,α,β< λn,δ

,

which leads to

ν f ∈W_,rα,β:f–Tnfq,α,β> λn,δ

≥ν f∈W_,r_α,β:QNf–TnQNfq,α,β> cmρλn,δ

.

LetLN:RN→AN andJN:AN→RN be defined by

LN(a)(x) = N

i=

aiϕi(x) ϕi,α,β

, a= (a, . . . ,aN)∈RN

and

JN(Fa) =

aϕ,α,β, . . . ,aNϕN,α,β

We see at once thatLNJN(Fa) =Fa for anyFa∈AN. Sety= (y, . . . ,yN)∈RN, whereyj=



ϕj,α,βf,ϕ

(ρ)

j . Theny=JNQN(f). Thus by (.) andϕj,α,βm–



, we obtain

LN(a)_q,α,βm–



q+_a

lNq. (.)

Combining (.) with (.), we conclude that for anyf ∈W_,rα,β,

QN(f) –TNQN(f)_q,α,βPN

QN(f)

–PNTnQN(f)Q_q,α,β

= LNJNQN(f) –LNJNPNTNLNJNQN(f)_q,α,β

m–q+_J

NQN(f) –JNPNTnLNJNQN(f)_lN q

m–q+_y–J

NPNTnLNylNq.

Remark thatgk= ϕk

ϕk,α,β,k= , , . . . ,N, is an orthonormal system inL,α,βandgk∈H(v) =

W_,ρ_α,β. Then the random vector (f,g_(ρ) , . . . ,f,g_N(ρ) ) =yinRN _{on the measurable space}

(W_,rα,β,ν) has the standard Gaussian distributionrNinRN. It then follows that

ν f ∈Wr

,α,β:QN(f) –TnQN(f)_q,α,β> cm ρ_λ

n,δ

≥ν f∈W_,rα,β:y–TJNPNTnLNylNq >cm ρ+q–_λ

n,δ

=rN y∈RN:y–TJNPNTnLNylNq >cm ρ+q–_λ

n,δ

=:rN(G),

wherecis a positive constant. Clearly,rank(JNPNTnLN)≤nand

rN(G)≤ν f ∈W,rα,β:f –Tnfq,α,β> λn,δ

≤δ.

Consequently,

λn,δ

IN :RN→lNq,rN

= inf

G infIN _x∈sup_RN_\G

INx–Tnx_lN q

≤ sup y∈RN_\G

INy–JNPNTnLNylNq

mρ+q–_λ

n,δ,

which implies

λn,δ

W_,rα,β,ν,Lq,α,β

m–ρ–q+_λ

n,δ

IN :RN→lNq,rN

n–ρ–q+_λ

n,δ

IN :RN→lNq,rN

.

This completes the proof of Theorem ..

Proof For the lower estimates, using Theorem . and Lemma ., we have for ≤q≤

λn,δ

Wr

,α,β,ν,Lq,α,β

n–ρ+/–/q_λ n,δ

IN:RN→lNq,γN

n–ρ+/–/q

N/q+N/q–/

ln



δ

/

n/–ρ

 +n–/

ln



δ

/

.

For ≤q<∞, we have

λn,δ

W_,rα,β,ν,Lq,α,β

n–ρ+/–/q

n/q+

ln



δ

/

n/–ρ

 +n–/q

ln



δ

/

.

And forq=∞,

λn,δ

W_,rα,β,ν,Lq,α,β

n–ρ+/–/q

lnm+ln



δ

/

= n/–ρ

ln

m

δ

/

.

It remains to prove the upper estimates. For ≤q≤ ∞and any fixed natural numbern, assumeCm≤n≤CmwithC>  to be specified later. We may take sufficiently small

positive numbersε>  such thatρ>_+ ( +ε)(max{α,β}+  +ε)(_–_q). Set

nj=

⎧ ⎨ ⎩

j+_{, ifj≤m,}

j+_(+ε)(m–j)–_{, ifj>m,}

and

δj=

⎧ ⎨ ⎩

, ifj≤m,

δm–j_{, ifj>m.}

Then

j≥

nj

j≤m

j+

j>m

m(+ε)–εjm

and

j≥

δj=δ

j≤m

m–j≤δ.

Thus, we can takeCsufficiently large so that

∞

j=

It follows from Lemma . forτ∈(,_{(max{α,β}}_{+)(/–/q)}), ≤q≤ ∞,

n

j=

b–_jτ(/–/q)k[+τ(/–/q)]_{= }k+kτ(/–/q)_.

Ifj≤m, thennj= j+, and thenceλnj,δj(Vj:R j+

→lj+

q ,γj+) = . Ifj>m, then taking 

τ = (max{α,β}+  +ε)(/ – /q) and applying Lemma ., Theorem ., we obtain for

≤q<∞,

λnj,δj

Vj:R

j+

→l_qj+,γ_j+

C(m,τ)

nj+ 

/τ

(j+)/q+ ln

δ

j(/–/q)_–(+ε)(m–j)(max{α,β}++ε)(_–_q)

qj _{+ ln}

δ

,

which yields

λn,δ

W_,rα,β,ν,Lq,α,β

∞

j=m+

–jρj(/–/q)–(+ε)(m–j)(max{α,β}++ε)(–q)_/–/q



j

q _{+ ln}

δ

–m(ρ–+q)

mq _{+ ln}

δ

n/–ρ

 +n–/q ln

δ

. (.)

Forq=∞, by Lemma . we get

λnj,δj

Vj:R

j+

→l_qj+,γ_j+

C(j+,τ) nj+ 

/τ

lnj+_+ln

δ

= j/–(+ε)(m–j)(max{α,β}++ε)/ j+ln

δ,

then applying Theorem ., we obtain

λn,δ

W_,r_α,β,ν,L∞,α,β

∞

j=m+

–jρ_j/–(+ε)(m–j)(max{α,β}++ε)/ _j+ln

δ

–m(ρ–) _m+ln δ n

/–ρ _lnn

δ. (.)

To finish the proof of the upper estimates, we only need to show that, for ≤q< ,

λn,δ

W_,rα,β,ν,Lq,α,β

λn,δ

W_,rα,β,ν,L,α,β

n/–ρ

 +n– ln

δ

/

.

5 Conclusions

In this paper, optimal estimates of the probabilistic linear (n,δ)-widths of the weighted Sobolev spaceWr

,α,βon [–, ] are established. This kind of estimates play an important

role in the widths theory and have a wide range of applications in the approximation the-ory of functions, numerical solutions of differential and integral equations, and statistical estimates.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Project no. 11401520), by the Research Award Fund for Outstanding Young and Middle-aged Scientists of Shandong Province (BS2014SF019), by the National Natural Science Foundation of China (11271263), by the Beijing Natural Science Foundation (1132001), and BCMIIS.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 6 March 2017 Accepted: 11 October 2017

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