Korovkin theorem in symmetric spacesAbstract. We give a simple proof of the following theorem: the set |1, sinx, cosx] is an

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVII (1987)

Lec h Ma l ig r a n d a (Poznan)

Korovkin theorem in symmetric spaces

Abstract. We give a simple proof of the following theorem: the set |1, sin x, cosx] is an У'4'-Korovkin set in a symmetric space X 2„ eln t if and only if X 2n is separable.

Let i f be a class of linear bounded operators on a Banach space X. A subset К a X is an ^-Korovkin set irr X if the conditions

supH T JI*^ ^ C for some C > 0,

n

and

lim \\T„k — k\\x = 0 for each k s K ,

П GO

imply that

lim \\T „ f-f\\x = 0 for each f e X .

n~>ao

If X is a Banach lattice, let i f + denote the positive operators on X.

P. P. Korovkin [4] showed that {1, a, a2] and {1, sin a, cos a] are i ^ + - Korovkin sets in C [0, 1] and C2n, respectively. Korovkin sets have also been studied in the //-spaces, primarily by V. K. Dzjadyk [2]. It is known that {1, a, a2} and {1, sin a, cos a} are i (?+-Korovkin sets in Lp[0, 1] and H2n (1 ^ p < oo), respectively. Berens and Lorentz [1] gave a generalization of Dzjadyk theorem for separable Banach function spaces containing L®. In this paper we shall give a simple proof of the above theorem and we prove that separability is necessary.

We consider only the periodic case. Let us denote by L°2n the F -space of all equivalence classes of Lebesgue measurable and 2rc-periodic functions on R, equipped with topology of convergence in measure. We will say that a Banach space X 2n is a Banach function space (on [0, 2л:]) if X 2n is a Banach subspace of L°2jt having the property that if / e l 2„ and g e l f 2n are such that

\g\ ^ | / | a.e. on [0, 2я], then g e X .2n and \\g\\Xln < \\f\\x2n-

AMS (MOS) subject classifications (1980), 41A36, 46E30.

K ey w o r d s an d p h r a s e s . Approximation by positive operators, symmetric spaces, convolution operators.

A Banach function space X 2n is said to be symmetric if f e X 2n and 9 e L°2j[ and \g\ is equimeasurable with |/ |, then g e X 2n and \\g\\x2n = \\f\\x2n- We denote by Int the set of all symmetric and interpolation spaces between l}2n and

^2п-TOO

1, sin x, and cos x will denote the three functions of C2n defined by 1 (f)

= l,'(sinx)(f) = sint, and (cosx)(t) = cos t for each t.

Th e o r e m 1 . ( a ) I f X 2n is a Banach function space containing and such that C2n is dense in X 2n, then [1, sinx, cosxj is an -Korovkin set in X 2n.

(b) I f Jl, sin x, cosx] is an J f +-Korovkin set in a symmetric space A ^ e l n t, then Х 2л is a separable space.

P ro o f, (a) Let / gX 2n and e > 0. Our objective is to show that there exist positive constants C x, C2, C3 and C4 such that

IITH f - f \ \ X2n ^ C x г + C2 1| Tn 1 - l\\x2n+ C 3 || Tn(sin x) - sin х||*2я

+ C4 1| 7^,(cos x ) - c o s x ||X2n.

If this is done, then our hypothesis implies that \\Tnf —f\ \x 2n < C5e must hold for all sufficiently large n. Therefore, it establishes that {1, sinx, cosx}

is an J ^ +-Korovkin set in X 2n.

To this end, start by observing that for each t the function 0 ^ g , e X 2n defined by gt (s) = sin2 [(s — r)/2] satisfies

2gt = 1 —cos (x — t) = 1 — sin t sinx — cost cosx.

Since each T„ is linear, we have

2(7^0,)(f) = LT„1 — sin t 7^(sinx) —cos r 7^, (cos x)](r)

= (T„l — 1) (r) — sinf [7^(sinx) — sinx] (t) — cos t [ Tn (cos x) — cos x] (t).

Hence

(1) 2||Т„0(||Х2я ^ \\T„\ — 1||лг211 + II7"„(sinx) — sin х||Х2я + 1|7^(cos x) — cosх||Х2я.

Since C2n is dense in X 2n, it follows that there exists a function h e C 2n such that

(2) \\f- h \\ x 2n<e. '

By the uniform continuity of h on [0, 2л:], there exists some Ô > 0 such that

\h(s) — h(t)\ < £ holds whenever s, r e [ 0 , 2л] satisfy \s — t\ <ô.

Now set M = sup \h(s)\. For all s, fG[0, 2л] we have (cf. [4]) 0^s^2n

\h{s)-h{t)\ <£ + —■2 M-21— gt (s).

sin

Namely, if |s —1\ < Ô, then (3) follows from \h(s) — h(t)\ < e and gt ^ 0, (3)

and if Ô ^ s — t < 2л —"<5, then

\h(s)-h(t)\ ^ 2 M < 2M sin2 9ÀS).

Hence, if s e (t — ô, 2л + t —<5], then inequality (3) holds. Since gt, h e C 2n, we have that

\h{s±2n)-h{t)\ < e + —.2 , -gt (s±2n)2 M sin \o

for s ± 2 n e ( ± 2 n + t — Ô, ± 2 n + 2 n + t — <5].

The proof of inequality (3) is now complete.

Since each Tn is positive, and linear, it follows from (3) that,

(4)

sin2^d T„gt ^ Tnh — h(t) Tn 1 < zTn 1 + 2 M sin2 ^<5Tn9t- Hence

(5) \\Tnh - h T nl\\x ^ e \ \ T nl\\x^ + ^ ~ - \ \ T ngt\\x

2n 2 Sin iu ²ⁿ

Thus, taking into account (1), we get from (2) and (5)

II/-

II/ -

II

1

1

1||*2я

+ • i ' i  \\Tngt\\X2 M 2n + C£ ^ e + M ||l — Tn l\\x +eCC' Sin 2.0

M ,

+ sj-n₂T ^ (II Tn1 - 4 x 2n + II Tn (sin x) - sin x\\x2n+\\T„ (cos x ) - c o s *||*2),

where C is the norm of the inclusion c I 2„.

The proof of the first part of the theorem is complete.

(b) We shall give an example of the sequence of convolution operators for which the Korovkin theorem cannot be applied in the case of arbitrary symmetric space X 2n.

Let / e l}2„, and, for each integer n, let Snf denote the nth partial sum of the Fourier series o f /. The Fejér sums o f / are-defined as

anf =S p f + S xf + . . . + S J ' n+ 1

<j„ is a convolution operator with the positive and 2K-periodic Fejér kernel 1 /sin [(//+ l)/2] t \2

K n{t)

2 ( /H - 1 )tc\ sin (f/2)

'and ||K„||Li n = 1.

We have IK 1 - 1||х2л = 0, lim Ж (sin x ) - s m x \ \ x 2n lim

n+ 1sinx

•sinx||; 2n 0,,and

lim 11 cr„ (cos x) — cosxll* = lim ---cos x —cos x

и + 1 их

0.

2n

The proof of Theorem 1 (b) follows from the following two lemmas.

Here l [ab] denotes the characteristic function of the interval [a, b~\.

Lemma 2. Let X 2n be an interpolation space between l}2n and with

2 л

interpolation constant one. The convolution integral operator (Tf)(t) = [ K(t о

— s) f (s) ds with a positive and In-periodic kernel К is bounded in the symmetric space X 2n if and only if the kernel К is integrable.

Moreover,

(6) 2“ 1 ||K||Li < \ \ T \ \x , .- x „ < \\K \ \Li 2n

P ro o f. For any a, 0 < a ^ 2n

a a a

^K(u)du = a ~ 1 K(u)du)ds ^ a~ 1 \( [ K(u)du)ds о о

2a 2n

a 2a —s

J( .f 0 —s

2 a

= a 1 j ( J K i t - s ) l [0.a]{s)ds)dt = a 1 f { T ll0^){t)dt

0 0 ^{0 .}

(by the Holder inequality)

^ a 1 II 7T[0,а ] llx2JI^[0,2a]llx'2n

{X'2n denotes associated .space to Х 2л)

^ a 1 ||Л1х2л-Х2л11 ^[0,а]Их2л ll^[0,2a]llx'2„ ^ 2 || Л1х2л-Х2л- On the other hand, since T is a convolution operator, we have

\\Tf\\Li < ||X||Li H/ILi and \\Tf\\Lao ^ P | | Ll H / i u .

2 n 2 n 2n 2n 2n 2 я

Hence, by interpolation, we have :

\\T f\\x?n < ii^ iL i im ix , This establishes the result.

We note that both the separable and rearrangement invariant spaces (rearrangement invariant space is a symmetric space whose norm has Fatou’s property or, equivalently, X 2n is isometric to X 2n) are interpolation spaces between I}2n and with interpolation constant one (the Calderôn-Mitjagin theorem; see [5], p. 142).

Le m m a 3. Let X 2n be a symmetric space. We have lim \\(7„f—f\ \x 2n = 0

n -* CO

for each f e X 2n if and only if X 2n is separable.

P ro o f. The fact that onf -* f i n the X 2n norm for each f e X 2n and the fact that onf is a trigonometric polynomial imply that trigonometric polynomials are dense in X 2n. Hence X 2n is separable.

On the other hand, if X 2n is a separable symmetric space, then trigonometric polynomials are dense in X 2n. Hence, if / e X 2n and f. > 0 are given, then there exists a trigonometric polynomial p such that \\f— p\\x2n <

We have

\W nf~f\\x2n ^ \№nf-GnP\\x2n+\WnP-p\\x2n+ \\p - f \\x 2n

< \\°п \\х 2п^ Х 2п\ \ / ~ P\\x2n+ C \ \ ° п Р ~ P\\Lf n+ \\p - f \\x 2n (from Lemma 2 and Fejér theorem)

^ \\f~P\\x2n + EX\\p—f l \x 2n <

for sufficiently large n and the lemma is proved.

The next corollary follows directly from Theorem 1.

Co r o l l a r y 4. The set [1, sinx, cosx} is an L£+-Korovkin set in a symmetric space X 2Ke l n t if and only if X 2n is separable.

It is therefore natural to ask:

Does the norm of convolution operator T in Lemma 2 is equal to ||K||Lin?

Is sup Цо'„11лг2п—x2jt finite for symmetric space Z 2n^Int?

References

[1] H. B e r e n s and G. G. L o r e n tz , Theorems o f Korovkin type for positive linear operators on Banach lattices, in: Approximation Theory, G. G. L o r e n tz , ed., New York 1973, 1-30.

[2] V. K. D z ja d y k , Approximation of functions by positive linear operators arid singular integrals (in Russian), Mat. Sb. 70 (1966), 508-517.

[3] V. К. D z ja d y k , An introduction to the theory of uniform approximation of functions by polynomials (in Russian), Moscow 1977.

[4] P. P. K o r o v k in , Linear operators and approximation theory (in Russian), Moscow 1959.

[5] S. G. K r e in , Ju. I. P e t u n in and E. M. S e m e n o v , Interpolation o f linear operators (in Russian), Moscow 1978.

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK POZNAN