R E S E A R C H
Open Access
Estimates of probabilistic widths of the
diagonal operator of finite-dimensional sets
with the Gaussian measure
Jiehua Zhou
1and Yuewu Li
2**Correspondence:
2School of Mathematical Sciences,
Hulunbeier University, Hulunbeier, Inner Mongolia 021008, China Full list of author information is available at the end of the article
Abstract
In this paper, we estimate the asymptotic orders of probabilistic and average widths of the compact embedding operators from the Sobolev spaceWr
2(T) intoLq(T)
(1≤q≤ ∞) with the Gaussian measure.
MSC: 41A10; 41A46; 42A61; 46C99
Keywords: probabilistic width; average widths; Sobolev space; Gaussian measure
1 Introduction and main results
Problems ofn-widths in the approximation theory have by now been studied in depth. A great deal of classical problems have been solved, and interesting new developments have appeared. For example, the problems of probabilistic, average and stochastic widths, which can reflect the behavior of function on the whole class and give information about the measure of the elements in the class that can be approximated to this or that de-gree, are the problems of this kind. For the results related to the probabilistic, aver-age and stochastic widths, the reader may be referred to Sul’din [, ], Traubet al.[], Maiorov [–], Mathé [–], Sun [, ], and Ritter []. The new developments in this direction can be found in Fang’s papers [–]. Moreover, Carl and Pajor [] proved an inequality with respect to the Gelfand numbers of an operatoru from N into a Hilbert space, from which one can immediately derive the inequality related to the Kolmogorov numbers by the known duality. In this article we continue the previous works and prove the estimates of probabilistic widths of the diagonal operators fromRm ontomq.
First, we recall some useful concepts. LetW be a bounded subset of a normed linear spaceXwith the norm · X, andFN be anN-dimensional subspace ofX. The quantity
e(W,FN,X) =sup x∈W
e(x,FN,X),
where
e(x,FN,X) = inf y∈FN
x–yX
is called the deviation ofWfromFN. It shows how well the ‘worst’ elements ofWcan be approximated byFN. The number
dN(W,X) =inf FN
e(W,FN,X) =inf FN
sup
x∈W
inf
y∈FN
x–yX,
whereFN runs through all possible linear subspaces ofXof dimension at mostN, is called the Kolmogorov’sN-width ofWinX. Assume thatWcontains a Borel fieldBconsisting of open subsets ofWand equipped with a probabilistic measureμdefined onB. That is, μis aσ-additive nonnegative function onB, andμ(W) = . Letδ∈[, ] be an arbitrary number. The corresponding probabilistic Kolmogorov’s (N,δ)-width of a setW with a measureμin the spaceXis defined by
dN,δ(W,μ,X) =inf Gδ
dN(W\G,X), ()
whereGδ runs through all possible subsets inBwith measureμ(Gδ)≤δ. Thep-average Kolmogorov’sN-width is defined by
dN(a)(W,μ,X)p=inf FN
W
e(x,FN,X)pdμ(x)
/p
, <p<∞, ()
whereFN in () runs over all linear subspaces ofXof dimension at mostN. Letmp be an
m-dimensional normed space of vectorsx= (x, . . . ,xm)∈Rm, with a norm
xmp = ⎧ ⎨ ⎩
(mi=|xi|p)/p, ≤p<∞,
max≤i≤m|xi|, p=∞.
Consider inRmthe standard Gaussian measurev=v
m, which is defined as
v(G) = (π)–m/
G
exp
– x
dx,
whereGis any Borel subset inRm. Obviously,v(Rm) = . Denote byBm
p(ρ) ={x∈mp :xp≤ρ}the ball of radiusρinmp. LetBmp =Bmp(). Let N = , , . . . , δ ∈[, ) be arbitrary and Tm be a linear invertible operator from
Rmontom
q. We define the probabilistic (N,δ)-width of the operator acting in spaceRm equipped with the Gaussian measurevinmq-norm:
dN,δ Tm:Rm→mq,v
=inf
G infLN
e Tm Rm\G
,LN,mq
,
wherev(G) <δ,dimLN≤N.
Maiorov in [] proved the following result.
Theorem A[] For m> N,δ∈(, /], ≤q≤,then
dN,δ Rm,mq,v
mq–m+ln(/δ).
Theorem B[] Let T be an operator fromm into a Hilbert space H.Then
dN(T)≤C
log
(mN + )
N /
T
for≤N≤m,m= , , . . . ,where C> is a universal constant.
Detailed facts about the usual widths, such as the Kolmogorov’s N-widths andNth Gelfand numbers (or GelfandN-widths) ofTwere given in the books [–].
Remark
(a) Theorem A shows the asymptotic expression of the probabilistic widths of the identity embedding fromRmintom
q,≤q≤.
(b) Theorem B gives the upper estimate of Gelfand numbers of operators fromm into a Hilbert space, and some of its striking applications in the geometry of Banach spaces and Rademacher processes can be found in []. By the dual relation, it is easy to obtain the similar upper estimate of the Kolmogorov’sN-widthsdN(T)of operators fromm intom∞,i.e.,
dN(T)≤C
log
(mN + )
N /
T.
(c) Motivated by Theorems A and B, in general cases, here we investigate the
asymptotic estimate of probabilistic widths for diagonal operators fromRmontom q, ≤q≤ ∞.
Now we are in a position to formulate our main results.
Theorem For m>N,δ∈(, /],then
dN,δ Tm:Rm→m∞,v
≤CTm + (/N)ln(/δ)
ln(em/N).
Theorem For m> N,δ∈(, /],then
dN,δ Tm:Rm→m,v
≥CTm
m+ln(/δ).
2 Proof of main results
In order to prove Theorems and , we also need some auxiliary assertions.
Lemma Letδ∈(,√/eπ],and let Tmbe a bounded linear invertible operator fromRm
ontom∞.Then,for any vector z∈Rm,
v x:(Tmx,z)≥Tm
ln(/δ)z
≤δ.
Proof First, assume thatTm is a diagonal operator of Rm, i.e., Tmx= (λixi)mi=, for any
known that
Tm= max
≤i≤m|λi|=|λ|.
Sincevis invariant with respect to orthogonal transformation ofRm, it suffices to prove the lemma for the vectorz∗= (z, , . . . , ). Letε∈(,e–] be arbitrary. We have
v x: Tmx,z∗≥ Tm
ln(/ε)z
=vx:λxz≥ |λ|
ln(/ε)z
=vx:|x| ≥
ln(/ε)
=√ π
∞
√
(/)ln(/ε)
exp –tdt≤
ε π
/
. ()
Here we use the inequality
∞
u
exp –tdt<
uexp–u
, u≥√ .
From () we obtain the assertion of the lemma byδ=√ε/π.
Next, assume thatTmis a symmetric transformation ofRm, then there is an orthogonal matrixUof ordermsuch that the matrixUTmUTis a diagonal matrix. Since the Gaus-sian measure is invariant for orthogonal transformation, the result holds for symmetric transformationTm.
Finally, assume thatTm is a general invertible linear transformation fromRmontoRm, then there are two matricesUandSsuch thatTm=US, whereUis an orthogonal matrix andSis a positive definite symmetric matrix. As the same reason above, the result holds for the transformationTm.
Thus Lemma is proved.
The following inequality will be used (see []). For any integersNandmwithm>N≥
, there exists a subspaceHofRmof dimensiondimH≥m–Nsuch that for anyx∈H,
x≤c
ln(em/N)
N /
x, ()
wherecis an absolute constant.
LetG⊂Rmbe a set. We introduce inRmanother norm for the operatorT
m:Rm→m∞:
xG= sup y∈Rm\G
(Tmy,x).
Lemma For anyδ∈(, /]and an arbitrary operator TmfromRmontom∞,there exists
a subset G=GδofRmwith measure v(G)≤δsuch that
sup
z∈Bm
∩H
zG≤cTm
Nln
exp(aN) δ ln
em N
/ ,
Proof Letk= [(N+ )/ln(em/N)] and consider thek–-net
S=(s/k, . . . ,sm/k) :s, . . . ,sm∈Z,|s|+· · ·+|sm| ≤k
for theBm inm
∞-norm. Using the inequality m
≤(em ), we estimate the cardinality ofS:
cardS≤
k
=
m+
≤
k
=
em
≤
em k
k
≤eaN, ()
whereais some absolute constant.
Consider the polyhedronQ=Bm ∩k–Bm∞. LetQ be the set of extremal points ofQ. The setQ consists of vectors withkcoordinates equal to±k–and the remaining coordinates zero. This implies thatQ ⊂S, and hencecardQ ≤exp(aN).
Letε=δ/exp(aN). InRmwe consider the setG=
s∈SGs, where
Gs=
y∈Rm:(Tmy,s)≥Tm
ln(/ε)s
.
Letz∈Bm
∩Hbe any point, ands∈Sbe a point closest tozinm∞-norm. Thenz=s+t
for somet∈Q. From Lemma ,
zG≤ sG+tG≤Tm
ln(/ε)s+tG. ()
Using the definition ofQ, we havet
≤ tt∞≤k–. From this and the definition of
Q andG,
tG≤max
t∈Q tG≤tmax∈S∩QtG
≤Tm
ln(/ε)max
t∈S∩Qt≤Tm
k–ln(/ε).
Therefore from (),
zG≤Tm
ln(/ε) z+t
+tG
≤Tm
ln(/ε) z+ k–/
. ()
Sincez∈H, it follows from the inequalities () and () that
zG≤cTm
k–ln(/ε)≤c Tm
(/N)ln(/ε)ln(em/N).
Using Lemma and the inequality (), we can estimate the measure ofG:
v(G)≤ s∈S
v(Gs)≤εcardS≤εexp(aN) =δ.
Proof of Theorem Using the duality inRmand Lemma , we have
sup
x∈Rm\G inf
y∈H⊥
Tmx–y∞= sup
x∈Rm\G sup
z∈H∩Bm
(Tmx,z)
= sup
z∈H∩Bm
zG≤cTm
(/N)ln(/ε)ln(em/N)
=cTm
(/N)ln exp(aN)/δln(em/N),
whereH⊥is the orthogonal complement ofHanddimH⊥≤N. The proof of Theorem
is completed.
Let us proceed to the proof of Theorem . For this, we first prove four lemmas. We introduce a definition. For arbitraryε> , theε-cardinalityof a subsetKofm is defined to be
Nε(K) =minN :z, . . . ,zN ∈Rm,e K,{z, . . . ,zN}
≤ε,
where
e K,{z, . . . ,zN}
=sup
x∈K
min
i=,...,Nx–zi
is the deviation ofKfrom the set{z, . . . ,zN}inm.
Letλbe a Lebesgue measure inR, normalized by the conditionλ(B) = , whereB=Bm . We consider Kolmogorov’s (N,δ)-width of the ballBwith measureλin them-norm:
dN,δ≡dN,δ Tm:B→m,λ
=inf
G infL e Tm(B\G),L, m
,
where theTmis as above, the infima are over all possible subsetsG⊂Bof measureλ(G)≤ δand all subspacesL⊂RmwithdimL≤N.
Lemma Suppose that D⊂B is an arbitrary subset with measureλ(D)≤δ.Then,for any
ε>dN,δ,
Nε Tm(B\D)
≤
+ Tm ε–dN,δ
m
.
Proof Leth> be any number, and letHbe any subspace ofRwithdimH≤Nsuch that
e Tm(B\D),H,m
–h≤dN,δ. ()
Letε =ε–dN,δ. We consider the setQ= (Tm(B\D))∩H. LetQε ={z, . . . ,zN}be the maximal subset ofQsuch thatzi–zj≥ε for alli=j. Clearly, by maximalityQε is a ε-net ofQfor · . The ballszi+ (ε/)Bm are disjoint and all contained inQ+ (ε/)Bm. Therefore, taking volumes we can obtain
N
i=
vol zi+ ε/
Hence, we have
N · ε/vol Bm≤vol Q+ ε/Bm. ()
ByQ= (Tm(B\D))∩H⊂Tm(B)⊂Tm(Bm )⊂Tm(Bm) and (), we have
N ε/mvol Bm≤ Tm+ ε/
m
vol Bm,
that is,
N ≤
+Tm ε
m
. ()
Now, we need to establishe(Tm(B\D),{z, . . . ,zN})≤ε. Sincet∈Q⊂H, we have from ()
sup
x∈Tm(B\D)
min
t∈Qx–t≤x∈Tsupm(B\D)mint∈Qhmin∈H
x–h+h–t
≤ sup
x∈Tm(B\D) min
t∈Qx–h+x∈Tsup
m(B\D) min
t∈Qhmin∈H
h–t
≤dN,δ+h. ()
From the inequality (), the definition ofQε and (), it follows that
e Tm(B\D),{z, . . . ,zN}
= sup
x∈Tm(B\D)
min
i=,...,Nx–zi
≤ sup
x∈Tm(B\D)
min
i=,...,Nmint∈Q x–t+t–zi
≤ sup
x∈Tm(B\D)
min
t∈Q
x–t+ min
i=,...,Nt–zi
= sup
x∈Tm(B\D) min
t∈Qx–t+ε
≤dN,δ+h+ε
=ε+h.
Consequently, lettingh→, we get thate(Tm(B\D),{z, . . . ,zN})≤ε, which together with
() completes the proof of Lemma .
From the relation (see [])
vol Bmp= (/p+ )m/ (m/p+ ), ≤p≤ ∞,
the ballsBmp satisfy the inequalities
cpm–m/p<vol Bmp< c pm–m/p, ()
where is the Euler -function, andcp,c pdepend only onp.
Lemma If Tmis a diagonal operator fromRmontom,then
det(Tm)=
m
i= λi(Tm)
Tm
√ m
m
,
whereλi(Tm),i= , . . . ,m,are non-zero eigenvalues of the operator Tmrearranged as usual
so that|λi(Tm)|is non-increasing and each eigenvalue is repeated according to its
multi-plicity.
Proof It is known that
Tm=
m
i= λi(Tm)
/ .
Accordingly,
√
mλm(Tm)≤ Tm ≤
√
mλ(Tm).
Obviously,
λm(Tm) m
≤
m
i= λi(Tm)
≤λm(T) m
,
from which the result of Lemma follows immediately.
Lemma Ifδ∈[, ]andλ(D)≤δ,then
Nε Tm(B\D)
≥
( –δ)cTm/ε
m .
Proof We first establish the inequality
Nε Tm(B\D)
≥
λ(Tm(B\D))
λ(Bm(ε)) . ()
Indeed, suppose that () does not hold. Then, for N =Nε(Tm(B\D)) and some set of pointsz∗, . . . ,z∗N, by
ε≥ sup
x∈Tm(B\D) min
i=,...,N
x–z∗i
≥
λ(Tm(B\D))
Tm(B\D)
min
i
x–z∗iλ(dx)
≥
λ(Tm(B\D))
Tm(B\D)\iN=(z∗i+Bm(ε)) min
i x–z
∗
iλ(dx)
≥
λ(Tm(B\D)) (ε)λ
Tm(B\D)
N
i=
z∗i +Bm(ε)
= ε
–N· λ(B
m (ε)) λ(Tm(B\D))
≥ε· =
ε,
we have obtained a contradiction.
In the sequel, we may as well assume thatTmis a diagonal operator fromRmontom. Using the inequality (), () and Lemma , we have
Nε Tm(B\D)
≥
λ(Tm(B\D)) λ(Bm(ε))
=
|det(Tm)|λ(B\D) λ(Bm
(ε))
≥
c
Tm
√ m
m
( –δ) vol(B)
vol(Bm (ε))
≥
c√Tm
m
m
( –δ)
c
√ m
ε
m
= ( –δ)
cTm
ε
m
.
Next, assume thatTmis a symmetric transformation ofRm, then there is an orthogonal matrixUof ordermsuch that the matrixUTmUTis a diagonal matrix. Since the Lebesgue measure is invariant for orthogonal transformation, the result holds for symmetric trans-formationTm.
Finally, in the general case,Tmis a general invertible linear transformation fromRmonto m , then there are two matricesUandSsuch thatTm=US, whereU is an orthogonal matrix andSis a positive definite symmetric matrix. As the same reason above, the result holds for the transformationTm.
Thus, we complete the proof of Lemma .
Lemma Ifδ∈[, /],then
dN,δ Tm:B→m ,λ
≥cTm.
Proof From Lemma and Lemma , we get
c
+ Tm ε–dN,δ
m
≥Nε Tm(B\D)
≥
( –δ)
cTm
ε
m
≥
cTm
ε
m
. ()
Letε= dN,δ. Taking the logarithm of the inequality (), we get the inequality
dN,δ Tm:B→m ,λ
≥cTm
for some constantscandcandNwithm≥cN. Lemma is proved.
Proof of Theorem According to Lemma , for anyδ∈[, /] and any subspaceL⊂Rm withdimL≤N, there is a setK⊂Bwith Lebesgue measureλ(K) >δsuch that
e Tmx,L,m
for any elementx∈K. On the unit sphereSm–={x∈Rm:x
= }, we consider the subsetK ={x/x:x∈K}.
LetλSm– be a Lebesgue measure on the sphereSm–. We prove that
λSm– K>δλSm– Sm–. ()
Indeed, assume that
λSm– K
≤δλSm– Sm–
.
We introduce inRm a polar system of coordinates (r,s), wherer≥ ands∈Sm–, and consider inBthe cone
C=(r,s) : ≤r≤,s∈K.
Then
λ(K)≤λ(C) =
vol(B)
rm–dr
K
λSm–(ds)
≤ δ
vol(B)
rm–dr
Sm–
λSm–(ds) =δ.
We have obtained a contradiction.
Consider the setKt={(r,s) :r≥t,s∈K},t≥. Using the inequality (), we estimate the Gaussian measure ofKt:
v(Kt) = (π)–m/
Kt exp
– u
(du)
= (π)–m/
∞
t
rm–exp
–r
dr
K
λSm–(ds)
≥(π)–m/δ
∞
t
rm–exp
–r
dr
Sm–
λSm–(ds)
=δvx:x≥t
. ()
A direct computation shows that fort≥√m,
v x≥t
≥exp –t and vx≥
√
m≥c> ,
wherecis some absolute constant. It follows from this and () that fort=max{√m,
√
ln(/δ)}and for anyδ∈(,c],
v(Kt)≥δ·vx>max √
m,ln(/δ)>δ. ()
For any elementy=rs∈Kt, we have from ()
e Tmy,L,m
=re Tms,L,m
≥crTm ≥cTmmax
√
m,ln(/δ)
≥ c Tm
SinceLis an arbitrary subspace withdimL≤N, it follows from () and () that
dN,δ Tm:Rm→m ,v
≥c Tm
m+ln(/δ).
Theorem is a direct consequence of this.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors completed the paper together. They also read and approved the final manuscript.
Author details
1Hulunbeier Vocational and Technological College, Hulunbeier, Inner Mongolia 021000, China.2School of Mathematical
Sciences, Hulunbeier University, Hulunbeier, Inner Mongolia 021008, China.
Acknowledgements
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. The present investigations was supported by the Natural Science Foundation of Inner Mongolia Province of China under Grant 2011MS0103.
Received: 20 October 2012 Accepted: 15 May 2013 Published: 3 June 2013 References
1. Sul’din, P: Wiener measure and its applications to approximation theory I. Izv. Vysš. Uˇcebn. Zaved., Mat.6, 145-158 (1959) (in Russian)
2. Sul’din, P: Wiener measure and its applications to approximation theory II. Izv. Vysš. Uˇcebn. Zaved., Mat.5, 165-179 (1960) (in Russian)
3. Traub, JF, Wasilkowski, GW, Wo´zniakowski, H: Information-Based Complexity. Academic Press, New York (1988) 4. Maiorov, VE: Discretization of the problem of diameters. Usp. Mat. Nauk30, 179-180 (1975)
5. Maiorov, VE: On linear widths of Sobolev classes and chains of extremal subspaces. Mat. Sb.113(115), 437-463 (1980). English transl. in Math. USSR Sb.41(1982)
6. Maiorov, VE: Kolmogorov’s (N,δ)-widths of the spaces of the smooth functions. Russ. Acad. Sci. Sb. Math.79, 265-279 (1994)
7. Maiorov, VE: Linear widths of function spaces equipped with the Gaussian measure. J. Approx. Theory77, 74-88 (1994)
8. Mathé, P:S-Numbers in information-based complexity. J. Complex.6, 41-66 (1990) 9. Mathé, P: Random approximation of Sobolev embeddings. J. Complex.7, 261-281 (1991)
10. Mathé, P: A minimax principle for the optimal error of Monte Carlo methods. Constr. Approx.9, 23-29 (1993) 11. Mathé, P: On optimal random nets. J. Complex.9, 171-180 (1993)
12. Mathé, P: Approximation theory of stochastic numerical methods. Habilitationsschrift, Fachbereich Mathematik, Freie Universität Berlin, Berlin (1994)
13. Sun, Y, Wang, C:μ-Averagen-widths on the Wiener space. J. Complex.10, 428-436 (1994)
14. Sun, Y, Wang, C: Average error bounds of best approximation of continuous functions on the Wiener space. J. Complex.11, 74-104 (1995)
15. Ritter, K: Average-Case Analysis of Numerical Problems. Lecture Notes in Math., vol. 1733. Springer, New York (2000) 16. Fang, G, Qian, L: Approximation characteristics for diagonal operators in different computational settings. J. Approx.
Theory140, 178-190 (2006)
17. Fang, G, Qian, L: Optimization on class of operator equations in the probabilistic case setting. Sci. China Ser. A50, 100-104 (2007)
18. Fang, G, Ye, P: Probabilistic and average linear widths of Sobolev space with Gaussian measure. J. Complex.19, 73-84 (2003)
19. Fang, G, Ye, P: Probabilistic and average linear widths of Sobolev space with Gaussian measure inL∞norm. Constr. Approx.20, 159-172 (2004)
20. Chen, G, Fang, G: Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure. J. Complex.20, 858-875 (2004)
21. Chen, G, Fang, G: Linear widths of multivariate function spaces equipped with the Gaussian measure. J. Approx. Theory132, 77-96 (2005)
22. Carl, B, Pajor, A: Gelfand numbers of operators with values in a Hilbert space. Invent. Math.94, 479-504 (1988) 23. Pinkus, A:n-Widths in Approximation Theory. Springer, New York (1985)
24. Carl, B, Stephani, I: Entropy, Compactness and the Approximation of Operators. Cambridge University Press, Cambridge (1990)
25. Pietsch, A: Operator Ideals. North-Holland, Amsterdam (1980) 26. Pietsch, A: Eigenvalues ands-Numbers. Geest und Portig, Leipzig (1987)
27. Gluskin, ED: Norm of random matrices and widths of finite-dimensional sets. Mat. Sb.120(162), 180-189 (1983). English transl. in Math. USSR Sb.48(1984)
doi:10.1186/1029-242X-2013-277
