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
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 setF
there corresponds a bounded setr
of points inIR3 given byr:=
{(f(t),f'(t),f"(t)) I f E F, t EIR}. In this note the boundary ofr
ischaracterized 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.
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 best00 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 bound00 00 00
for II f' II is best possible. Also, if II f i l s 1 and II fIt II
s
1 onIR, then00 00 00
II f'lI
s
/2,
and this bound is best possible. Later (1939) Kolmogorov [1J com-00n 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 bySCh~enberg
and Cavaretta (cf. [4, p. 297-308J). For some special linear differential oper.ators Land L., the Landau problemn ~
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
- 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 (tan2)
+ M tan2
is minimal. A simple computation shows that if 0 ~ M < 1 then h=
~.
in case M ~ lone has2 '
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' II21M.
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 00II 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 • 002.2. 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 function4
-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 show00
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.
Thesere-00
sults imply the following theorem
e
Theorem 2.2. If II fll ::: 1 and II f" - f II ::: M on [0,(0), then00
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 :::
~
+ a00 00 00 a
21M (M
~
a2).00
References
e
[1J Kolmogorov, A.N., On inequalities between the upperbounds of thesucces-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