An extremum problem of the Landay type concerning the differential operators $D%5E2\pm I$ on the halfline

An extremum problem of the Landay type concerning the

differential operators $D%5E2\pm I$ on the halfline

Citation for published version (APA):

Morsche, ter, H. G. (1979). An extremum problem of the Landay type concerning the differential operators $D%5E2\pm I$ on the halfline. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7906). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979 Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 79-06 August 1979

An extremum problem of the Landau type concerning 2

the differential operators D ±I on the half line

Technological University Department of Mathematics PO Box 513, Eindhoven The Netherlands.

by

Abstract

An extremum problem of the Lnndau type concerning

2

the differential operators D ±I on the halfline

by

H.G. ter Morsche

Let f: [0,00) ~ R be bounded and let f have an absolutely continuous

deriva-tive fl. In this note the following result is established:

i f Ilfll ₀₀ ~ 1 and IIf" + fll ₀₀ ~ M on [0,00), then

II f 11 ₀₀ ~ M + 1 (0 ~ M ~ 1) and

II fl II ₀₀ ~

21M

(M ~ 1) •

Moreover, these bounds are best possible for II f I II •

00

- 2

-1. Introduction

In 1974 Schoenberg (cf. [3, p. 353-369J) pro~ed the following results. Let

f : lR + a: be bounded and let f have a absolutely continuous derivat ive f' • Then one has: a) if II f 1100 = 1 and II f" + f II ~ M, then II f I II ~

12M

+ 1, and this bound is best

00 00

possible for II f I II ,

00

b) if ll fll = l and ll f"-fll ~M, then ll f'lI ~ M(O ~ M~l)and

00 00 00

II f I 1100 ~ 12M - 1 (M > 1), and again these bounds are best possible for II f I 11

00•

These results can be considered as solutions of extremum problems of the

Lan-~ dau type, problems that can be formulated in a more general form as follows. Let f be defined in a closed interval of the real axis and let f have an

ab-solutely continuous (n - l)-th derivative f(n-1) . Let Land L. be linear dif

-n ~

ferential operators with constant coefficient of order nand i respectively,

wi th n > i. I f II f II ~ 1 and II L f II ~ M then the problem is to compute the best

00 n 00

constants M. (i

=

1, ••. ,n-l) in the inequalities IIL.fll ~ M. (i

=

1, .•• ,n-1) .

~ ~ 00 J.

The first results go back to 1913, when Landau [2J proved the following

in-equalities: if II fll ~ 1 and II filII ~ 4 on [0,00), then II f' II ~ 4, and this bound

00 00 00

for IIf' II ₀₀ is best possible. Also, if IIf ll ₀₀ ~ 1 and II f" II ~ 1 onIR, then

00

II fill ~

12,

and this bound is best possible. Later (1939) Kolmogorov [lJ

com-00

n i d

puted the best constants in the case L

=

0 , L.

=

0 , where 0

= --

and the

n J. dx

interval considered is the whole real axis.

The much more difficult case L

=

on, L

=

oi but now considered on the half -n

~

line was solved in 1970 by Schoenberg and Cavaretta (cf. [4, p. 297-308J). For some special linear differential operators Land L., the Landau problem

n J.

on the real axis is studied by Sharma and ~zimbalario [5J.

2 In this note we derive the best constants for the Landau problem, L2

=

0 ±I, L1

=

0 on the halfline.

2. Computation of the best constants

~.1. We first consider the differential operator 02 + I. Let f be defined on [0,00)

and let f have an absolutely continuous derivative fl. Assuming h E (O,n) we

easily verify by partial integration that the following differentiation for-mula holds (2. 1) f' (x) where 1jJ (t) 1 . h(f(x + h) - cosh f(x» -s~n sin(h - t) sin h h

f

1jJ(t) (f"(x + t) + f(x + t»dt,

o

- 3

-I f II f II ~ 1 and II fn + fll ~ M i t follows from (2.1) that

00 00

(2.2) II f' II ~ 1 + cosh + 1 - cosh M = 1 M tan(h/2) (0 < h ~ ~2)

00 sinh sinh tan (h/2)

n h -1 h .

Now we choose h E (0'-2J such that (tan -) + M tan - 1S minimal. A simple

2 2

computation shows that if 0 ~ M < 1 then h

=

~

;

in case M ~ lone has h = 2 arctan (1/IM) .

Hence IIf'lI ~ M + 1 i f 0 ~ M < 1 and IIf' II ~ 21M if M ~ 1. The following two

00 00

examples show that these constants are best possible. If 0 ~ M < 1 we define f(x) f(x) -M + (M + l)sin x 1 (x > n/2) . (0 ~ x ~ n/2) ,

It follows that II fll ₀₀ = 1, II fn + f II ₀₀ = M , II f' II 00 M + 1. In case M ~ 1 we define f(x)

f(x)

-M + (M - l)cos x + 21M sin x (0 ~ x ~ 2 arctan 1/1M)

1 (x > 2 arctan 1/IM) .

Then it is easily verified that II f II = 1, II fn + f II

00 00 M, II f' II 00

21M.

Consequently, we have proved the following theorem.

Theorem 2. 1. I f II f II ~ 1 and II fn + f II ~ M on [ 0 , 0 0 ) , then

00 00

II f' II ₀₀ ~ M + 1 (0 ~ M < 1) and II f'

II

~ 21M (M ~ 1) •

00

e

Moreover , these bounds are best possible for II f' II • 00

2.2. Now we shall show that the result of theorem 2.1 also holds for the differen-2

tial operator D - I. In this case we use the differentiation formula

(2.3) where f' (x)

₌

_{sinh(h) (f(x + h)} 1 _{- cosh(h)f(x»} -I/J(t)

=

sinh(h - t) sinh (h) (h > 0) • h

f

I/J (t) (fn (x + t) - f(x +t) )dt

o

If II f IL, ~ 1 and II fn - f 1100 ~ M, and taking into account the identity cosh2(h) - sinh2(h)

=

1, we obtain

(2.4)

II

f' II ~ sinh (h) cosh(h) - 1 00 cosh(h) - 1 + M sinh (h)

We first consider the case 0 ~ M < 1. Since (2.4) holds for all h > 0 we ob-tain II f' 1100 ~ M + 1 by taking h -+ 00. We further observe that the function

4

-f(x)

=

-M + (M + 1) e -x (x ~ 0) has the properties

II

fll 00

₌

1,

II

f" - fll 00

=

M,

II

f 11100

=

M + 1- On the other hand, ifM ~ 1 then h > 0 is chosen such that cosh(h)

-

1

=

(1/rM) sinh (h) • As a consequence we have

II

fill ~

21M.

To show

00 that this bound for

II

f'

II

₀₀ is best possible we take the function

f (x) = M - (M + 1)cosh(x) + 21M sinh(x) (0 ~ x < h)

f(x) = 1 (x ~ h)

One easily verifies that

II

f

II

= 1,

II

f" - f

II

00 00 M and

II

fill 00

=

21M.

These re-sults imply the following theorem

e

Theorem 2.2. I f

II

fll ~ 1 and

II

f" - f

II

~ M on [0,(0), then 00

II

f'

II

₀₀ ~ _{M + 1 (0} ~ M < 1 ) and

II

fill ₀₀ ~ 21M (M ~ 1 )

.

Moreover, these bounds are best possible for

II

f I

II •

00

Remark. As an immediate consequence of theorems 2.1 and 2.2 we obtain a gene-ralisation of our results for the differential operators D2±a2I (a > 0). In-deed, let f be such that on [0,(0)

II

f

II

!'> 1 and

II

f"±a2f

II

~

M. Putting

00 00

g(x) := f(x/a) we obtain (0

~

M

~

a 2) and

II

f I

II

!'>

00

II

g" ±g

II

~

M/ a 2 and

II

gil !'> 1. Hence

II

f I

II

~!:!.

+ a

00 00 00 a

2rM (M

~

a2).

References

e

[1J _{Kolmogorov, A.N., On inequalities between the upperbounds of the}

succes-sive derivatives of an arbitrary function on an infinite interval. Amer. Math. Soc. Transl., Series 1, vol. 2 (1962), 233-243.

[2J Landau, E., Einige Ungleichungen fur zweimal differentierbare Funktionen. Proc. London Math. Soc.

! l

(1913), 43-49.

[3J Karlin, S., C.A. Micchelli, A. Pinkus and I.J. Schoenberg, Studies in spline functions and approximation theory. Academic Press Inc., New York-San Francisco-London, 1976.

[4J Penkov, B. and D. Va~ov, Constructive function theory. Publishing house of the Bulgarian academy of sciences, Sofia, 1972.

[5J Sharma, A. and J. Tzimbalario, Landau type inequalities for some linear differential operators. Illinois J. Math. 20 (1976), 443-455.