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The Landau problem for the differential operator $ D%5E3$

Citation for published version (APA):

Morsche, ter, H. G. (1979). The Landau problem for the differential operator $ D%5E3$. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7907). Technische Hogeschool Eindhoven.

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

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.

-EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 79-07 August 1979

The Landau problem for the differential operator D3

Technological University Department of Mathematics PO Box 513, Eindhoven The Netherlands. by H.G. ter Morsche

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Abstract

3

The Landau problem for the differential operator D

by

H.G. ter Morsche

The set of all real-valued functions f defined onIR having absolutely conti-1

nuous second derivatives and satisfying IIfll ~ ,lIf"lli ~ 1 is denoted by

00 24 00

F.

To the set

F

there corresponds a bounded set

r

of points inIR3 given by

r:=

{(f(t),f'(t),f"(t)) I f E F, t EIR}. In this note the boundary of

r

is

characterized by using cubic Euler splines. These spline functions play an important role in inequalities between derivatives of a function, also known as inequalities of the Landau type.

(4)

2

-1. Introduction

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

f : lR -+

a:

be bounded and let f have a absolutely continuous derivative f' • Then one has: a) if Ilflloo = 1 and IIf" + fll ~ M, then IIf'lI ~

12M

+ 1, and this bound is best

00 00

possible for II f' II ,

00

b) ifllfll =1andllf"-fll 00 00 ~M,thenllf'lI ~M(0~M~1)and 00

II f' II ~ 12M - 1 (M > 1), and again these bounds are best possible for II f' II •

00 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 - 1)-th derivative f(n-1). Let Ln and Li be linear dif-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-1) in the inequalities IIL.fll ~ M. (i

=

1, •.• ,n-1).

~ ~ 00 ~

The first results go back to 1913, when Landau [2J proved the following in-equalities: if II fll ~ 1 and II f"l1 ~ 4 on [0,(0), then II f' II

s

4, and this bound

00 00 00

for II f' II is best possible. Also, if II f i l s 1 and II fIt II

s

1 onIR, then

00 00 00

II f'lI

s

/2,

and this bound is best possible. Later (1939) Kolmogorov [1J com-00

n i d

puted the best constants in the case L = D , L. = D , where D = - - and the

n ~ dx

interval considered is the whole real axis.

The much more difficult case L = Dn, L = Di but now considered on the

half-~line

was solved in 1970 by

SCh~enberg

and Cavaretta (cf. [4, p. 297-308J). For some special linear differential oper.ators Land L., the Landau problem

n ~

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

2

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

=

D ±I, L1

=

D on the halfline.

2. Computation of the best constants

2.1. We first consider the differential operator D2 + I. Let f be defined on [0,(0) and let f have an absolutely continuous derivative f'. 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

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- 3

-If II f II 00 ~ 1 and II f" + fll 00 ~ M it follows from (2.1) that

(2.2) IIf't ~ 1 + cosh + 1 - cosh M sinh sinh

=

tan (n/2) 1 M tan(h/2) (0 < h ~ ~2)

~ h -1 h

Now we choose h E

(0'2J

such that (tan

2)

+ M tan

2

is minimal. A simple computation shows that if 0 ~ M < 1 then h

=

~

.

in case M ~ lone has

2 '

h

=

2 arctan(l/IiM).

Hence IIf' II 00 ~ M + 1 i f 0 ~ M < 1 and IIf' II 00 :c::; 21iM if M ~ 1. The following two examples show that these constants are best possible.

If 0 :c::; M < 1 we define f(x) f(x) -M + (M + l)sin x 1 (x > ~/2) • (0 :c::; x :c::; ~/2) ,

It follows that IIfll 00

=

1, IIf" + fll 00

=

M, IIf' II 00 M + 1. In case M ~ 1 we define f(x)

f(x)

-M + (M - l)cos x + 21iM sin x

1 (x > 2 arctan 1/1iM) .

(0 :c::; x :c::; 2 arctan 1/1iM)

Then it is easily verified that II f II

=

1, II f" + f II

=

M, II f' II

21M.

00 00 00

Consequently, we have proved the following theorem.

Theorem 2.1. If

II

fll 00 :c::; 1 and II f" + f II ~ M on [0,00), then 00

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

00 21iM (M ~ 1)

.

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 I f II f II 00 ~ 2 cosh (h) (2.4) h f' (x) = sinh (h) (f(x + h) - cosh(h)f(x» 1 -

f

1/1 (t) (f" (x + t) - f(x +t) )dt 1/I(t) = sinh(h - t) sinh (h) 1 and II f" - f 1100 :c::; - sinh2(h) = 1, we II f I 1100 :c::; sinh (h) cosh(h)

-o

(h > 0) •

M, and taking into account the identity obtain

cosh(h) - 1 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

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4

-f(x)

=

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

=

1, iJf" - fll

=

M,

00 00

II f' II 00 = M + 1. On the other hand, if M ~ 1 then h > 0 is chosen such that

cosh (h) - 1 = (l/IM) sinh (h). As a consequence we have II f I II :::

21M.

To show

00

that this bound for IIf'll is best possible we take the function

00

f(x) M - (M

+

l)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 f

I II =

21M.

These

re-00

sults imply the following theorem

e

Theorem 2.2. If 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' 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 [0,00) II f II 00 ::: 1 and II f"±a2f II 00 ::: M. Putting

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

(0 ::: M ::: a 2 ) and II f I II :::

IIg"±gll ::: M/a 2 and IIgll ::: 1. Hence IIf' II :::

~

+ a

00 00 00 a

21M (M

~

a2).

00

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. ~ (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

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