Lethargic Approximation: Banach and Fréchet Spaces Forming Bernstein Pairs（AKSOY）

Lethargic Approximation: Banach and Fréchet
Spaces Forming Bernstein Pairs
Asuman Güven AKSOYAbstract
In this paper, we examine the Bernstein’s Lethargy Theorem in the context of Banach and Fréchetspaces and deﬁne Bernstein pair. We introduce conditions under which a pair of Banach spaces
form a Bernstein pair. Some open questions relating Bernstein pairs for Fréchet spaces are alsopresented.
Mathematics Subject Classiﬁcation (2020): 41A25, 41A50, 41A65, 46B99.
Key words: Bernstein Lethargy Theorem, Bernstein Pairs, Approximation Numbers.
1

Introduction

One of the notable theorems used in the constructive theory of functions is Bernstein Lethargy
Theorem (BLT) [7] and this paper is motivated by this theorem. For f ∈ C([0, 1]), the sequenceof best approximation is deﬁned asρ(f, Pn ) = inf{||f − p|| : p ∈ Pn } for all n ≥ 1,

(1)

where Pn denotes the space of all polynomials of degree ≤ n. Clearly
ρ(f, P1 ) ≥ ρ(f, P2 ) ≥ · · ·and {ρ(f, Pn )} form a non-increasing sequence of numbers. S. N. Bernstein in [7] proved that if{dn }n≥1 is a non-increasing null sequence (i.e., lim dn = 0) of positive numbers, then there existsn→∞a function f ∈ C[0, 1] such that
ρ(f, Pn ) = dn for all n ≥ 1.
This remarkable result is called Bernstein’s Lethargy Theorem (BLT) and is used in the constructive theory of functions [24], and it has been applied to the theory of quasi analytic functions inseveral complex variables [22]. Note that the density of polynomials in C[0, 1] (the Weierstrass
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研究紀要

第 98 号

Approximation Theorem) implies thatlim ρ(f, Pn ) = 0.

n→∞However, the Weierstrass Approximation Theorem gives no information about the speed of convergence for ρ(f, Pn ). Bernstein’s Lethargy Theorem has been extended replacing C[0, 1] by arbitrary
Banach space X and replacing Pn to arbitrary closed subspaces of X [9]. Furthermore, Tyuriemskih [26] showed that the sequence of errors of best approximation from x to Yn , {ρ(x, Yn )}, mayconverge to zero at an arbitrarily slow rate. More precisely, for any expanding sequence {Yn } of

subspaces of X and for any null sequence {dn } of positive numbers, he constructed an elementx ∈ X such thatlim ρ(x, Yn ) = 0, and ρ(x, Yn ) ≥ dn for all n ≥ 1.

n→∞

However, it is also possible that the errors of best approximation {ρ(x, Yn )} may converge to zeroarbitrarily fast, for results of this type see [5].

We refer the reader to [11] for an application of Tyuriemskih’s Theorem to convergence of sequenceof bounded linear operators.
We also refer to [1, 3, 5, 16] for other versions of Bernstein’s Lethargy Theorem and to [2, 4, 17,
21, 28] for Bernstein’s Lethargy Theorem for Fréchet spaces.
Given an arbitrary Banach space X, a strictly increasing sequence {Yn } of subspaces of X anda non-increasing null sequence {dn } of non-negative numbers, one can ask the question whether

there exists x ∈ X such that ρ(x, Yn ) = dn for each n. For a long time no sequence {dn } of thistype was known for which such an element x exists for all possible Banach spaces X. The only
known spaces X in which the answer is always “yes” are the Hilbert spaces (see [27]). For a general(separable) Banach space X, a solution x is known to exist whenever all Yn are ﬁnite-dimensional(see [25]). Moreover, it is known that if X has the above property, then it is reﬂexive (see [27]).
2

Banach Spaces Forming Bernstein Pairs

Before we deﬁne the concept of Bernstein Pair, we need the deﬁne the approximation number.
Deﬁnition 1 Let X and Y be Banach spaces. For every operator T ∈ L(X, Y ) the n-th approximation number of T is deﬁned asan (T ) = inf{||T − S|| : S ∈ L(X, Y ), rank S < n}.
Clearly,−8−

Lethargic Approximation: Banach and Fréchet Spaces Forming Bernstein Pairs（AKSOY）||T || = a1 (T ) ≥ a2 (T ) ≥ · · · ≥ 0andan (T ) = ρ(T, Fn ),where Fn is the space of bounded linear operators from X to Y with rank at most n. Approximation
numbers are the largest s-numbers [20]. For operators on Hilbert spaces there are some well known

properties of an (T ). Namely, if X and Y are Hilbert spaces, then (an (T )) is the sequence of singularnumbers of T and moreover if T is a compact operator, then an (T ) = λn (T ∗ T )1/2 , where
λ1 (T ∗ T ) ≥ λ2 (T ∗ T ) ≥ · · · ≥ 0
are the eigenvalues of T ∗ T ordered as above. We refer the reader to [10, 19] for general information
about approximation and other s-numbers. We say T is approximable if lim an (T ) = 0. Anyn→∞approximable operator is compact, but the converse is not true due to the existence of Banach
spaces without approximation property.
Deﬁnition 2 Two Banach spaces X and Y are said to form a Bernstein pair (BP) if for any positive monotonic null sequence (dn ) there is an operator T ∈ L(X, Y ) and a constant M depending

only on T and (dn ) such thatdn ≤ an (T ) ≤ M dn for all n.
In this case we say that (an (T )) is equivalent to (dn ) and write (X, Y ) to denote Bernstein pair,
and in case M = 1, (X, Y ) is called an exact Bernstein pair.
Here by an (T ) we mean the n-th approximation number of T deﬁned above.
Theorem 3 For any Hilbert space H, (H, H) form an exact Bernstein pair.
Proof. If both spaces X and Y are equal to a Hilbert space H and if {dn } is a sequence decreasing

to 0, then there exists a compact operator T on H such thatan (T ) = dn(see [20]). Thus (H, H) form an exact Bernstein pair.

All of these desirable properties of an (T ) for operators between Banach spaces brings us to thefollowing question:
Question: Suppose {dn }n≥1 is a non-increasing null sequence (i.e., dn  0+ ) of positive numbers.
Does there exist T ∈ L(X, Y ) such that the sequence (an (T )) behaves like the sequence {dn }?

In other words, suppose dn  0+ ,

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研究紀要

第 98 号a) Does there exist T ∈ L(X, Y ) such that an (T ) ≥ dn for any n?

b) Does there exist T ∈ L(X, Y ) such that an (T ) = dn for any n?
c) Does there exist T ∈ L(X, Y ) and a constant M such thatdn
≤ an (T ) ≤ M dn
M
for any n?
First results of this kind appeared in [14] in the context of Banach spaces with basis. They proved
the following theorem.
Theorem 4 (C. Hutton, J. Morrell and J. Retherford [14]) Suppose {dn }n≥1 is a nonincreasing null sequence of positive numbers and X is a Banach space with basis. Then there
exists T ∈ B(X) such thatdm ≤ am (T ) ≤ Kdm ,where the constant Kdepends on X.
Later these results sharpened as follows:
Theorem 5 (A. G. Aksoy and G. Lewicki [3]) Suppose {dn }n≥1 is a non-increasing null sequence of positive numbers. In each of the following cases there exists T ∈ B(X, Y ) such thatam (T ) = dm

for each ma) X and Y has 1-unconditional basis (for example X = Y = ℓp for 1 ≤ p < ∞ and X = Y =c0 ).

b) X ∈ {c0 , ℓ∞ } and Y has 1-symmetric basis.
c) Y = ℓ1 and X has 1-symmetric basis.

In [3] among other things, it was proved that if we assume (X, Y ) is a Bernstein pair with respectto {an } and suppose that a Banach space W contains an isomorphic and a complementary copyof X, and a Banach space V contains an isomorphic copy of Y , then (W, V ) is a Bernstein Pairwith respect to {an }. This implies that there are some natural pairs of Banach spaces that forma Bernstein pair as shown in the following.

Theorem 6 For 1 < p < ∞ and 1 ≤ q < ∞, the couple (Lp [0, 1], Lq [0, 1]) form a Bernstein

Pair.

Proof. This follows from the fact that (ℓ2 , ℓ2 ) is a Bernstein pair with respect to {an } ([19])and the fact that for every p, 1 ≤ p < ∞, Lp [0, 1] contains a subspace isomorphic to ℓ2 andcomplemented in Lp [0, 1] for p > 1 (see [29], page 85).

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Lethargic Approximation: Banach and Fréchet Spaces Forming Bernstein Pairs（AKSOY）

Example 7a) If (ℓ∞ , ℓ∞ ) is BP, then (X, Y ) is BP provided that both X and Y contains isomorphic copyof ℓ∞ (Phillips Theorem).
b) If (c0 , c0 ) is a BP, then (X, Y ) is BP provided that both X and Y each contain an isomorphic
copy of c0 and X is separable (Sobczyk’s Theorem).
c) If (ℓ1 , ℓ1 ) is a BP, then (X, Y ) is BP provided that both X and Y each contain an isomorphic
copy of ℓ1 and X is a non reﬂexive subspace of L1 [0, 1] (Pelczynski-Kadeč Theorem).
For the statements of Phillips, Sobczyk and Pelczynski-Kadeč Theorems we refer the reader to[12].
The most recent result on the rate of decay of approximation numbers can be found in [18], where
it is shown that for two inﬁnite dimensional Banach spaces X and Y and a non-increasing nullsequence {dn }n≥1 , there exists an approximable T : X → Y such that||T || ≤ 2d1

anddm /9 ≤ am (T ) ≤ 3d[m/4]for any m.

It is also worth mentioning that in [15] operators with prescribed eigenvalue sequences are constructed.
3Fréchet Spaces and Bernstein Pairs

In this section we deﬁne Fréchet spaces, mention recent results of Bernstein’s Lethargy Theorem forFréchet spaces and pose questions whether or not two Fréchet spaces X and Y form a Bernstein
pair. Fréchet spaces are locally convex spaces that are complete with respect to a translation
invariant metric and they are generalization of Banach spaces.
Deﬁnition 8 (X,  · F ) is called a Fréchet space, if it is a metric linear space which is completewith respect to its F-norm || · ||F giving the topology. As usual by a F-norm we mean that || · ||Fsatisﬁes the following conditions [23]:1. ||x||F = 0 if and only if x = 0,

2. ||αx||F = ||x||F for all real or complex α with ||α|| = 1,
3. ||x + y||F ≤ ||x||F + ||y||F .

Many Fréchet spaces X can also be constructed using a countable family of semi-norms ||x||k ,where X is a complete space with respect to this family of semi-norms. For example a translation
invariant complete metric inducing the topology on X can be deﬁned as− 11 −

研究紀要

d(x, y) :=

∞
∑

k=0

2−k

第 98 号||x − y||kfor x, y ∈ X.
1 + ||x − y||kClearly, every Banach space is a Fréchet space, and the other well known example of a Fréchetspace is the vector space C ∞ [0, 1] of all inﬁnitely diﬀerentiable functions f : [0, 1] → R, where thesemi-norm is||f ||k = sup{|f k (x)| : x ∈ [0, 1]}.
For more information about Frechét spaces the reader is referred to [23].
Bernstein’s Lethargy Theorem (BLT) is also studied for Fréchet spaces.
The most recent result in this direction can be found in [2]. More precisely, let X be an inﬁnitedimensional Fréchet space and let V = {Vn } be a nested sequence of subspaces of X such thatVn ⊆ Vn+1 for any n ∈ N. Let dn be a decreasing sequence of positive numbers tending to 0.

Under an additional natural condition on sup{dist(x, Vn )}, we proved that there exists x ∈ X andno ∈ N such thatdn
≤ ρ(x, Vn ) ≤ 3dn
3

for any n ≥ no .

Using the above-mentioned theorems of BLT type it is natural to examine Bernstein pairs in thecontext of Fréchet spaces. In other words we ask what are the conditions under which two Fréchetspaces X and Y to form a Bernstein pair? To be able to obtain results like Bernstein’s Lethargy

Theorem, one useful condition is to look at the deviation of the subset Vn from Vn+1 as deﬁned in
the following.
Notation 9 Let (X,  · F ) be a Fréchet space and assume that V = {Vn } is a nested sequence of

linear subspaces of X satisfying Vn ⊂ Vn+1 . Then the deviation of Vn from Vn+1 is deﬁned asdn,V = sup{ρ(v, Vn ) : v ∈ Vn+1 }.

(2)Here in this paper we assume thatdV = inf{dn,V : n ∈ N} > 0.

(3)

Necessity of this assumption is illustrated in the following example.
Example 10 Let X = {x = (xn ) : xn ∈ R for any n ∈ N} be equipped with the F -norm xF =
∞
∑
|xj |
. Let
j (1 + |x |)2
jj=1− 12 −

Lethargic Approximation: Banach and Fréchet Spaces Forming Bernstein Pairs（AKSOY）

Vn = {x ∈ X : xk = 0

for k ≥ n + 1}.

It is easy to see that for any x ∈ X
ρ(x, Vn ) =

∞
∑

1
|xj |
≤ n.
j (1 + |x |)
2
2
j
j=n+1

2
1
2
1
and observe that for any x ∈ X, ρ(x, Vn ) ≤ n < . Also observe that dn,V = n+1 ,n2
n2
which implies that dV = 0.

Let dn =Note that in case of Banach space X, we have dn,V = +∞ becauseρ(tx, Vn ) = tρ(x, Vn )and the supremum taking over all v ∈ Vn+1 and Vn is strictly included in Vn+1 .

The next example is to show that there is a natural way to think of Fréchet spaces when dn,ν = 1.

Example 11 Let (X,  · ) be a Banach space. Deﬁne in X an F-norm  · F by xF = x/(1 +x). Then dn,V = 1 for any n ∈ N independently of V. Because the mapping
t →

t
1+t

is increasing for t > −1 andρF (tx, Vn ) :=ρ(tx, Vn )→1
1 + ρ(tx, Vn )as t → ∞.

Theorem 12 ([2]) Let X be a Fréchet space and assume that V = {Vn } is a nested sequence of

linear subspaces of X satisfying Vn ⊆ Vn+1 , where the closure is taken with respect to  · F . Let

dn,V be deﬁned as above and {en } be a decreasing sequence of positive numbers satisfying
∞
∑

j=n

2j−n (δj + ej ) < min{dn,V , en−1 }with a ﬁxed sequence of positive numbers δj . Then, there exists x ∈ X such that ρ(x, Vv ) = dn forany n ∈ N.

Let X, Y be Fréchet spaces. Then, n-th approximation number of T : X → Y is deﬁned asan (T ) = inf{||T − S||F : S ∈ L(X, Y ), rank S < n}.
We again have
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研究紀要

第 98 号||T ||F = a1 (T ) ≥ a2 (T ) ≥ · · · ≥ 0.
In the light of the Theorem given in [14], we know that if X is a Banach space with basis, then(X, X) forms a Bernstein pair, thus it is relevant to identify those Fréchet spaces which have basis.
Theorem 13 Let X be a nuclear Fréchet space, then (X, X) forms a Bernstein Pair.
Proof. It is known that nuclear Fréchet spaces have basis (see [13])Questions:a) Let X and Y be Fréchet spaces, is it true that if we know that (X, Y ) is an exact Bernsteinpair with respect to {an } and if we suppose that a Fréchet space W contains an isomorphic

and complementary copy of X, and a Fréchet space V contains an isomorphic copy of Y ,then (W, V ) is a Bernstein pair with respect to {an }?

b) As for those Fréchet spaces which are not nuclear can we deﬁne Bernstein pairs and give
examples?
To answer part a) perhaps using diagonal operators examining a theorem of BLT type for the
space of bounded linear operators L(X, Y ), under the condition dν > 0, could be helpful.

For part b) recalling S. Bellenot’s paper(s) such as Local Reﬂexivity of Normed Spaces Operators
and Fréchet Spaces in [6], there he prove a non-uniformly convex Fréchet spaces has a conditional
basic sequence, could be used.
References[1] Aksoy A. G. and Almira J., On Shapiro’s lethargy theorem and some applications, Jaén J. Approx.,
6 (1), 87–116 (2014).
[2] Aksoy A. G. and Lewicki G., Bernstein’s lethargy theorem in Fréchet spaces, J. Approx. Theory, 209,
58–77 (2016).
[3] Aksoy A. G. and Lewicki G., Diagonal operators, s-numbers and Bernstein pairs, Note Mat., 17,
209–216 (1999).
[4] Albinus G., Remarks on a theorem of S. N. Bernstein, Studia Math., 38, 227–234 (1970).
[5] J. M. Almira and N. del Toro, Some remarks on negative results in approximation theory, in: Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory,
Vol. I (Potenza, 2000). Rend. Circ. Mat. Palermo (2) Suppl. 2002, no. 68, part I, 245–256.
[6] Bellenot, S. F., Local reﬂexivity of normed spaces, operators, and Fréchet spaces, J. Func. Anal. 59,
no.1, 1-11 (1984).
[7] Bernstein S. N., On the inverse problem in the theory of best approximation of continuous functions,Collected works (in Russian), vol II, Izd. Akad. Nauk, USSR, 292–294 (1954).
[8] Borodin P. A., Selected approximative properties of sets in Banach spaces, Ph.D. Dissertation,
Moscow State University, Moscow (2012).

− 14 −

Lethargic Approximation: Banach and Fréchet Spaces Forming Bernstein Pairs（AKSOY）[9] Borodin P. A., On the existence of an element with given deviations from an expanding system ofsubspaces, Mathematical Notes, 80 (5), 621–630 (translated from Matematicheskie Zametki 80 (5),657–667 (2006)) (2006).
[10] Carl, B and I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge
University Press, Cambridge, 1990.
[11] Deutsch F. and Hundal H., A generalization of Tyuriemskih’s lethargy theorem and some applications,
Numer. Func. Anal. Opt., 34 (9), 1033–1040 (2013)Sequences and Series in Banach Spaces, SpringerVerlag, Graduate text in Math., New York, 1983.
[12] Diestel, J. Sequences and Series in Banach Spaces, Springer-Verlag, 1984.
[13] Dubinsky, E. The structure of Nuclear Fréchet spaces, Springer Lecture Notes in Mathematics, No:
720, 1979.
[14] C. Hutton, J. S. Morrell and J. R. Retherford, Diagonal operators, approximation numbers andKolmogorow diameters, J. Approx. Theory, (16), 48-80 (1976).
[15] Kaiser, R. and J. Retherford, Eigenvalue distribution of nuclear operators: a survey in: Vector
Measures and Integral Representation of Operators, Essen University Press, Essen, 1983, pp.245-287.
[16] Lewicki G., A theorem of Bernstein’s type for linear projections, Univ. Iagellon. Acta Math., 27,
23–27 (1988).
[17] Micherda B., Bernstein’s lethargy theorem in SF-spaces, Z. Anal. Anwendungen, 22 (1), 3–16 (2003).
[18] T. Oikhberg, Rate of decay of s-numbers, J. Approx. Theory 163, (2011), 311-327.
[19] Pietch, A., Eigenvalues and s-Numbers, Cambridge University Press, Cambridge, 1987.
[20] Pietch, A., Operator Ideals, North Holland, Amsterdam, 1980.
[21] Pleśniak W., On a theorem of S. N. Bernstein in F -Spaces, Zeszyty Naukowe Uniwersytetu Jagiellonskiego, Prace Mat., 20, 7–16 (1979).
[22] Pleśniak W., Quasianalytic functions in the sense of Bernstein, Dissertations Math., 147, 1–70 (1977).
[23] Rolewicz, S., Metric Linear spaces, PWN, Warszawa, 1982.
[24] Singer I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. SpringerVerlag, Berlin (1970).
[25] Timan A. F., Theory of Approximation of Functions of a Real Variable. Dover publications, NewYork (1994).
[26] Tyuriemskih I. S., On one problem of S. N. Bernstein, in: Scientiﬁc Proceedings of Kaliningrad State
Pedagog. Inst., 52, 123–129 (1967).
[27] Tyuriemskih I. S., The B-Property of Hilbert Spaces, Uch. Zap. Kalinin. Gos. Pedagog. Inst., 39,
53–64 (1964).
[28] Vasil’ev A. I., The inverse problem in the theory of best approximation in F -spaces, (In Russian)Dokl. Ross. Akad. Nauk., 365 (5), 583–585 (1999).
[29] Wojtaszczyk, Banach Spaces for Analysts, Studies in Advanced Mathematics, vol. 25, Cambridge
University Press, Cambridge, 1991.
Asuman Güven Aksoy,
Crown Professor of Mathematics and Robert’s Fellow
Claremont McKenna College
Department of Mathematics
Claremont, CA, 91711, USA
E-mail: aaksoy@cmc.edu− 15 −