On the pointwise convergence of a family of functionals on C ( I )

of Functional Equations, Approximation and Convexity, Cluj-Napoca, May 21 – May 25, 2003, pp. 3–13.

On the pointwise convergence of a family of functionals on C ( I )

Mira-Cristiana Anisiu (Cluj-Napoca)

Valeriu Anisiu (Cluj-Napoca)

Abstract. Given a continuous functionu: [0,∞)→R, a family of function- alsϕα:C(I)→R,α >0, is defined byϕα(f) =_α¹

α

R

0

u(t)f(t/α) dt.It is proved that the necessary and sufficient conditions for the familyϕα, α >0 to satisfy

α→∞lim ϕα(f) =

α→∞lim

1 α

α

R

0

u(t)dt

·

1

R

0

f are:

I.∃ lim

α→∞

1 α

α

R

0

u(t)dt; II. sup

α>0 1 α

α

R

0

|u(t)|dt <∞.

Iff ∈ C¹(I),condition I alone implies the existence of lim

α→∞ϕα(f).

A sequence of functionals (ϕn)n∈Nis attached to a numerical sequence (an)n∈N

which is Ces`aro-convergent toa,namely ϕn(f) = 1

n

n

X

k=1

akf(k/n), f Riemann integrable.

Additional conditions are imposed on the sequence (an)n∈N in order to prove that

n→∞lim ϕn(f) =a· Z ¹

0

f.

MSC 2000: 47B38, 26E60

1 Introduction

For the intervalI = [0,1] and a Banach spaceF 6={0},let us denote by B(I, F) the Banach space of bounded functionsf :I → F endowed with the sup norm. The subspace ofB(I, F) of regular functions (which admit side limits at eacht∈I) will be denoted byR(I, F) ; the Banach space of continuous functions C(I, F) is a subspace of R(I, F).

Given a sequence of real numbers (a_n)_n∈N, a sequence of operators ϕ_n:R(I, F)→F, n∈N,

(1.1) ϕ_n(f) = 1

n

n

X

k=1

a_kf k

n

can be generated. It was proved in [1] that:

A. The operatorsϕ_n are linear and continuous.

B. If the numeric sequence satisfies the conditions:

B₁. (a_n)_n∈N is Ces`aro-convergent toa( lim

n→∞

a1+...+aⁿ n =a);

B2. the sequence

|a1|+...+|aⁿ| n

n∈N is bounded, then the sequence (ϕ_n(f))_n∈N is convergent and

(1.2) lim

n→∞ϕ_n(f) =a· Z ₁

0

f.

C. If lim

n→∞ϕn(f) exists for everyf ∈ C(I, F) ⊆ R(I, F),the condi- tionsB₁ and B₂ from above are also necessary.

D. Iff ∈ C¹(I, F) (i.e. f is continuous with a continuous derivative), the result in B holds even if conditionB₂ is omitted.

The aim of this paper is to provide continuous variants of these results for families of functionals (for the sake of simplicity we consider F =R).

Letu : [0,∞)→ Rbe a continuous function. We define a family of

functionals associated tou, namelyϕ_α :C(I)→R, α >0, given by (1.3) ϕα(f) = 1

α

α

Z

0

u(t)f t

α

dt, f ∈ C(I).

Proposition 1.1 For each α >0, the functional ϕ_α is linear and con- tinuous, and its norm is given by

(1.4) kϕ_αk= 1

α Zα

0

|u(t)|dt.

Proof. This result is classical; see for a simple proof [3]. It also holds ifC(I) is replaced with the Banach spaceR(I) of regular functions.

2 Main results for families of functionals

As mentioned in B and C in the introduction, in the discrete case the conditionsB1 and B2 are necessary and sufficient for the sequence (1.1) to converge, the limit being given by (1.2). We can prove a similar result for the continuous case.

Theorem 2.1 Let there be given f ∈ C(I) and u ∈ C([0,∞)). If the function u satisfies the conditions:

I. ∃ lim

α→∞

1 α

Rα 0

u(t)dt;

II.sup

α>0 1 α

Rα

0 |u(t)|dt <∞, there exists the limit lim

α→∞ϕ_α(f) and

(2.1) lim

α→∞ϕ_α(f) =



 lim

α→∞

1 α

Zα

0

u(t)dt



· Z1

0

f.

The conditions I and II are also necessary in order to have (2.1) for each f ∈ C(I).

Proof. Let us suppose that conditions I and II hold. We can prove (2.1) even for the more general case of a regular functionf. To this end it suffices to prove (2.1) for f ∈ E with E ⊆ R(I) and spE = R(I) (condition II allows then to obtain (2.1) for any f ∈ R(I)). As a setE we choose the set of characteristic functions χ_[a,b], [a, b]⊆I.

Forf =χ_[a,b] we have ϕ_α(f) =

1

R

0

u(αt)f(t)dt=

b

R

a

u(αt)dt= _α¹

αb

R

αa

u(t)dt

= b_αb¹ Rαb 0

u(t)dt−a_αa¹

αaR

0

u(t)dt.

It follows

αlim→∞ϕ_α(f) = (b−a) lim

α→∞

1 α

Rα 0

u(t)dt

=

αlim→∞

1 α

Rα 0

u(t)dt

· R1 0

f.

Let us suppose now that (2.1) takes place for each f ∈ C(I). For f(x) = 1, ∀x ∈ I we obtain condition I. To prove II, we apply the Banach-Steinhaus principle for sequences αn → ∞, because it cannot be used directly for generalized sequences. (In fact, if X is an infinite dimensional Banach space, then its dual X^∗ is sequentially closed and dense in X^#,weak^∗

, see [6, p. 138].) It follows that sup

α>0kϕαk < ∞, and using the expression ofkϕ_αk given in (1.4) we obtain II.

Remark 2.1 If the function u is periodic withu(t+T) =u(t) for each t >0, then

αlim→∞

1 α

Zα

0

u(t)dt= 1 T

ZT

0

u(t)dt

and condition II automatically holds because of the boundedness ofu. In

this special case we obtain

(2.2) lim

α→∞ϕ_α(f) = 1 T

ZT

0

u(t)dt· Z1

0

f,

which is a result due to L. Fej´er, see [4, p. 114].

Remark 2.2 Condition II does not follow from condition I. Indeed, let u∈ C([0,∞))be given by

u(x) =





 2√

n+ 1 (x−2n), x∈[2n,2n+ 1/2)

−2√

n+ 1 (x−2n−1), x∈[2n+ 1/2,2n+ 3/2) 2√

n+ 1 (x−2n−2), x∈[2n+ 3/2,2n+ 2) (n∈N). For this function we have

sup

α>0

1 α

α

Z

0

|u(t)|dt≥ sup

n∈N∗

1 2n

2n

Z

0

|u(t)|dt = sup

n∈N∗

1 2n

n

X

k=1

√k=∞.

For2n≤α <2n+ 2 it follows ¹_α Rα 0

u(t)dt= _α¹ Rα 2n

u(t)dt≤ ^√_2αⁿ⁺¹ ≤ ^√_4nⁿ⁺¹ and lim

α→∞

1 α

α

R

0

u(t)dt= 0.

If we consider only functions in C¹(I),we obtain the corresponding property as in D mentioned in the introduction for sequences of opera- tors.

Theorem 2.2 Let there be given a function f ∈ C¹(I) and u ∈ C([0,∞)). If the function u satisfies condition I from Theorem 2.1

∃ lim

α→∞

1 α

Rα 0

u(t)dt=a

, then there exists lim

α→∞ϕ_α(f) and its value is given by (2.1).

Proof. Let U be an antiderivative of u with U(0) = 0 (i.e. U(x) = Rx

0

u(t)dt). We have forf ∈ C¹(I)

ϕ_α(f) = _α¹

α

R

0

U⁰(t)f _α^t dt

= _α¹U(α)f(1)−_α¹² Rα 0

U(t)f⁰ _α^t dt

= _α¹U(α)f(1)−_α¹

1

R

0

U(αt)f⁰(t) dt.

But _α¹U(αt) = _α¹ Rαt 0

u(s)ds and we get lim

α→∞

1

αU(αt) = ta. Using the theorem of dominated convergence we get

αlim→∞ϕ_α(f) =af(1)−a Z1

0

tf⁰(t) dt=a· Z1

0

f.

3 A new result for sequences of operators

In connection with the result mentioned inB in the introduction for the sequence of operators ϕ_n : R(I, F) → F, n ∈ N, given by (1.1), an open question was formulated in [1]: Is the conclusion in B true if F =Rforf Riemann integrable (instead of regular)?

We shall prove that this result holds if the sequence (a_n)_n∈N is bounded from above or below, or if

|a1|+···+|aⁿ| n

n∈N is convergent.

Theorem 3.1 Let there be given a Riemann integrable function f :I → R and a sequence(a_n)_n∈N of real numbers satisfying the conditions:

1. lim

n→∞

a1+...+aⁿ

n =a;

2. the sequence

|a1|+...+|aⁿ| n

n∈N is bounded;

3. the sequence (a_n)_n∈N is bounded from above or from below, or |a1|+...+|aⁿ|

n

n∈N is convergent.

Then the sequence(ϕn(f))_n∈^N given by (1.1) is convergent toa· R1 0

f.

Proof. (i) Let us consider a sequence (a_n)_n∈N with a_n ≥ 0, which satisfies condition 1 (hence also condition 2). Given ε > 0, for the Riemann integrable functionf there exist the continuous functionsu, v: I →Rsuch that

(3.1) u≤f ≤v and

Z

(v−u)< ε.

Then the functionalsϕ_n given by (1.1) will satisfy

(3.2) ϕn(u)≤ϕn(f)≤ϕn(v).

From the result in [1] mentioned at B in the introduction we have

nlim→∞ϕn(v) = a· R1 0

v, hence there exists n1 ∈N so that for any n≥n1, ϕ_n(v)< a·

R1 0

v+ε. Condition (3.1) implies that R1 0

v <

R1 0

f+ε,hence

(3.3) ϕn(v)< a· Z1

0

f+ε(a+ 1).

Similarly, there exists n₂ ∈Nso that for any n≥n₂,

(3.4) a·

Z1

0

f−ε(a+ 1)< ϕ_n(u).

From (3.2), (3.3) and (3.4) we obtain

a· Z1

0

f−ε(a+ 1)< ϕ_n(f)< a· Z1

0

f +ε(a+ 1),

hence

ϕ_n(f)−a· R1 0

f

≤ε(a+ 1) for n≥max{n₁, n₂} and the conclu- sion holds.

(ii) Let us consider (an)_n∈^N which satisfies the conditions 1 and 2 and is bounded from below, i. e. a_n ≥ −α. The sequence (b_n)_n∈N, b_n=a_n+α is Ces`aro-convergent toa+α and has b_n≥0; applying the result proved in (i) it follows that

nlim→∞

1 n

n

X

k=1

(an+α)f k

n

= (a+α)·

1

Z

0

f,

hence lim

n→∞ϕn(f) =a· R1 0

f.

(iii) If the sequence (a_n)_n∈N satisfies 1 and 2, and is bounded from above (a_n≤α),the sequence (c_n)_n∈N, c_n=α−a_n is Ces`aro-convergent to α−a and has c_n ≥ 0. Applying again the result proved in (i) we obtain the conclusion.

(iv) Let us consider the sequence (a_n)_n∈N with lim

n→∞

a1+...+aⁿ

n = a

and lim

n→∞

|a1|+...+|aⁿ|

n =a^∗.We can write a_n= a_n+|a_n|

2 −|a_n| −a_n

2 .

The sequence

an+|an| 2

n∈N is Ces`aro-convergent to ^a+a₂^∗ and has non- negative terms, hence it follows from (i) that

nlim→∞

1 n

n

X

k=1

a_n+|a_n| 2

f

k n

= a+a^∗

2 ·

Z1

0

f.

Similarly, for

|aⁿ|−aⁿ 2

n∈N we have

nlim→∞

1 n

n

X

k=1

|an| −an

2

f k

n

= a^∗−a

2 ·

1

Z

0

f.

It follows

nlim→∞

1 n

n

X

k=1

a_nf k

n

= a+a^∗

2 ·

Z1

0

f−a^∗−a 2 ·

Z1

0

f =a· Z1

0

f.

Remark 3.1 The conditions in 3 of Theorem 3.1 are not consequences of 1 and 2, as the following examples show.

Example 3.1 A Ces`aro-convergent sequence (a_n)_n∈N which is not bounded from above or from below, for which

|a1|+...+|aⁿ| n

n∈N is bounded, is given by

a_n=







k, forn= 2^k, keven

−k, forn= 2^k, k odd 0, otherwise.

Example 3.2 A Ces`aro-convergent sequence (an)n∈N, for which |a1|+...+|aⁿ|

n

n∈N is bounded without being convergent, can be obtained from a bounded sequence dn ≥0 for which ^d1+...+d_n ⁿ does not converge, as

a_n=

d_n/2, forneven

−d_(n+1)/2, forn odd.

Such a sequence (d_n)_n∈N is, for example,

dn=







1, forn∈

2^k,2^k+1

, keven 0, forn∈

2^k,2^k+1

, k odd.

Indeed, we have

k→∞lim, k even

d₁+...+d₂^k+1₋₁

2^k+1−1

= lim

k→∞

2⁰+ 2²+...+ 2^k

2^k+1−1

= 2 3, and

k→∞lim, k odd

d₁+...+d₂^k+1₋₁

2^k+1−1

= lim

k→∞

2⁰+ 2²+...+ 2^k−1

2^k+1−1

= 1 3. Remark 3.2 We mention, in connection with Theorem 3.1, the follow- ing result from [2] (see also [5]):

Fork >0, if f, g: [0,∞)→R satisfy:

− f ∈ C¹([0,∞)) is decreasing with f(∞) = 0 and there exists

αlim→∞

Rα 0

f(x+kα)dx=a;

−g∈ C([0,∞))is bounded and ∃ lim

α→∞

1 α

Rα 0

g=b, then lim

α→∞

Rα 0

f(x+kα)g(x)dx=ab.

References

[1] M.-C. Anisiu, V. Anisiu. Sequences of linear operators related to Ces`aro-convergent sequences. Revue d’Analyse Num. Th. Approx.

31(2) (2002), 139-145.

[2] R. Gologan. On the convergence of some partial sums and a contest problem. Gazeta Matematic˘a seria A, Anul XIX nr. 2 (2001), 70-73.

[3] I. J Maddox, The norm of a linear functional. Amer. Math. Monthly 96(1989), 434-436.

[4] G. Sz´asz, L. Geh´er, I. Kov´acs, L. Pint´er (editors).Contests in Higher Mathematics. Akad´emiai Kiad´o, Budapest 1968.

[5] E. C. Titchmarch.The Theory of Functions. Oxford University Press 1939.

[6] A. Wilanski. Modern Methods in Topological Vector Spaces.

McGraw-Hill 1978.

T. Popoviciu Institute of Numerical Analysis 37, Republicii st., 3400 Cluj-Napoca

Romania

Babe¸s-Bolyai University, Faculty of Mathematics 1, Kog˘alniceanu st., 3400 Cluj-Napoca

Romania