internat. j. math. sci

Internat. J. Math. & Math. Sci.VOL. 18 NO. 3 (1995) 443-446

443

PROPERTIES OF THE MODULUS OF CONTINUITY FORMONOTONOUS CONVEX FUNCTIONS AND APPLICATIONS

SORIN GHEORGHE GAL

Department of MathematicsUniversity of Oradea

Str. Armatei Romane 53700 Oradea, Romania

(Received January 29, 1992 and in revised form November 22, 1992)

ABSTRACT. For a monotone convex function f e C[a,b] we prove that the modulus of

continuity w(f;h) is concave on [a,b] as function of h. Applications to approximation theory axe

obtained.

KEY WORDS AND PHRASES. Concave modulus of continuity, approximation by positive

linear operators, Jackson estimate in Korneichuk’s form.

1991 AMS SUBJECT CLASSIFICATION CODES. 41A10, 41A36, 41A17.

1. INTRODUCTION.In a recent paper, Gal [1] the modulus of continuity for convex functions is exactly

calculated, in the following way.

THEOREM 1. (see [1]) Let f 6 C[a,b] be monotone and convex on [a,b]. For any h [0,b-a]we have:

(i) w(f;h) f(b)-f(b-h), if f is increasing on [a,b],

(ii) w(f;h) f(a)- f(a + h), if/" is decreasing on [a,b],where w(f;h) denotes the classical modulus of continuity.

Denote

KM[a,b] {f C[a,b]; f is monotonous convex or monotonous concave on [a,b]}The purpose of the present paper is to prove that for f KM[a,b] the modulus of continuity

w(]’;h) is concave as function of h 6 [0,b-a] and to apply this result to approximation by positive

linear operators and to Jackson estimates in Korneichuk’s form.

2. MAIN RESULTS AND APPLICATIONS.A first main result is the followingTHEOREM 2. For all/" gM[a,b], the modulus of continuity w(f;h) is concave as function

of h [0, b- a].PROOF. Let firstly suppose that /’ is increasing and convex on [a,b]. If/’ is increasing on

[a,b], by Theorem 1, (i), we have w(f;h)= f(b)- f(b-h). Hence

andaw(f;hl) + (1 a)w(f; h2) f(b)-af(b- hl)- (1 -oOf(b- h2)

w(f;ch + (1 a)h2) f(b)-f(b-ohl-(1-oOh2)

for all c 6 [0,1] and all hl,h2 [0,b-a].

(1.1)

(1.2)

444 S.G. GAL

Since f is convex on [a,b] we get

f(b ah -(1 ()h2) < f(b hl + (1 a)f(b h2),

wherefrom taking into account (1.1) and (1.2) too, we get

a.(f:hl) + (l- a)w(f;h.2) _< w(f;t,h /(1- a)h2) (1.3)

Now, if f is decreasing on [a.b], since by Theorem 1, (ii), we have u(f.h)= f(a)-f(a+h), we

immediat el)" get

cw(f;hl)+ (1 -)(f;h2) f(a)-cf(a + hi)-(1 -()f(a + h2)and

u(f;ah + (1- a)h2) f(a)-f(a+ah + (1 a)h2)

for all c, e [0,1] and all hl,h2 e [0,b- a].Since f is convex on [a,b] we have

f(a + ah + (1 a)h2) < oI(a + hi) + (1 a)f(a + h2)

which together with (1.4) and (1.5) gives again (1.3).In the following we need the

DEFINITION 1. (see e.g. [2]) Let f C[a,b] be. If (f;h) sup{If(x)- f(Y) l; Ix- Y < h} is

the usual modulus of continuity, the least concave majorant of u(f;h) is given by

"(,;h)--sup{(6-)w(f;fl)+(-’)w(f;) O<o<6<<b_a}.An immediate consequence of Definition is the

COROLLARY 1. For any f KM[a,b] we have

" (f;h) w(f;h)

PROOF. Putting c in Definition we get

w(/; h) _< (f; h).

Then, taking into account Theorem 2, for 0 _< a _< 6 _< _< b- a we have

wherefrom passing to supremum, we immediately get

(f; 6)

_,;(f; 6),

which proves the corollary.REMARK. It is easy to see that Corollary remains valid for all f e_ C[a,b] having a concave

modulus of continuity (f; h).

Now, firstly we will apply the previous results to approximation by positive linear operators.

Thus, investigating the sequence of Lehnhoff polynomials in [3], Ln(f)(x), defined for

f e C[- 1,1], H.H. Gonska [2] proves that

Ln(f)(x)-f(x)l <_ f;V/l x2/n2 }

Taking now into account Corollary we immediately get the

MONOTONOUS CONVEX FUNCTIONS AND APPLICATIONS 445

COROLLARY 2. If f e KM[ 1, l] then for all e 1,1], N we have

I,.,,::>:.>-:(.>I :,._+

In the sme ppe, for f C[0,1], H.H. GonsR obtains estimates in tems of the modulus

(;h) in the pproimtion by the so-cIIe Shep operator, S(f),l S S. hen by

CooIlry an by Theorem 4.3 in [2] e immediately et the

COROLLARY . For all I RM[0,1] nd II n N e hve

s),(f)- f " f;ln(2n + 2)

( 1) 1<"<2S(f)- f 5 f;(n + 1)"-

Finally, we will apply our results to the following so-called Jackson estimate in Korneichuk’s

forln.

THEOM 3. (see e.g. [4], p. 147) For any f C[- ,] we have

E._ () (f; ),. ,, .,

where E(f) denotes the best approximation by polynomials of degree ! .Now, we will prove the

THEOM 4. If f C[-I,i] has a concave modulus of continuity w(f;h),h[O,2], then we

hv

Extending to [0,=] by taking w(f;h)=w(f;2),hE[2,=], obviously w remnsPROOF.concave on [0, r].Denote w(h) w(f;h),h [0, Tr] and

A {g C[- 1,1];w(g;h) <_ w(h),Vh [0, r]}.

Obviously f e Aw. Then by [5, Theorem 8 and Lemma 2, p. 122-123], as in the proof of Theorem

9, p. 123 in [5], there is g LiPM1 such that

IlY-all <1/2(L ) rM2

Now by Theorem V, (ii), in [4, p. 147], there is Pn- polynomial of degree <_ n- such that

Hence we get

IIg-Pn 111 < ’M

II/-P._I < IlY-oll + Ila-Pn_l -<1/2 (f; )

which proves the theorem.

REMARK. For f E KM[- 1,1], Theorem 4 remains valid.

446 S. G. GAL

REFERENCES

1. GAL, S.G., Calculus of the modulus of continuity for nonconcave functions andapplications, Calcolo 27 (1990), 195-202.

2. GONSKA, H.H., On approximation in spaces of continuous functions, Bull. Austral. Math.Soc. 28 (1983), 411-432.

3. LEHNHOFF, H.G., A simple proof of A.F. Timan’s theorem, J. Approx. Theory 38 (1983),172-176.

4. CHENEY, E.W., Introduction to Approximation Theory, McGraw-Hill Book Company,New York, 1966.

5. LORENTZ, G.G., Approximation of Functions, Holt, Rinehart and Winston, SecondEdition, New York, 1986.

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