The existence, characterization and essential uniqueness of solutions of 𝐿^{∞} extremal problems
Authors: Fisher S. D. , Jerome J. W. 
Published: July 2010

Let

I
=
(
a
,
b
)

I = (a,b)

be an interval in R and let

H

n
,
∞

{H^{n,\infty }}

consist of those real-valued functions f such that

f

(
n
−
1
)

{f^{(n - 1)}}

is absolutely continuous on I and

f

(
n
)

∈

L
∞

(
I
)

{f^{(n)}} \in {L^\infty }(I)

. Let L be a linear differential operator of order n with leading coefficient

1
,
a
=

x
1

>
⋯
>

x
m

=
b

1,a = {x_1} > \cdots > {x_m} = b

be a partition of I and let the linear functionals

L

i
j

{L_{ij}}

on

H

n
,
∞

{H^{n,\infty }}

be given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript i j Baseline f equals sigma-summation Underscript v equals 0 Overscript n minus 1 Endscripts a Subscript i j Superscript left-parenthesis v right-parenthesis Baseline f Superscript left-parenthesis v right-parenthesis Baseline left-parenthesis x Subscript i Baseline right-parenthesis comma j equals 1 comma midline-horizontal-ellipsis comma k Subscript i Baseline comma i equals 1 comma midline-horizontal-ellipsis comma m comma">

L

i
j

f
=

∑

v
=
0

n
−
1

a

i
j

(
v
)

f

(
v
)

(

x
i

)
,

j
=
1
,
⋯
,

k
i

,
i
=
1
,
⋯
,
m
,

{L_{ij}}f = \sum \limits _{v = 0}^{n - 1} {a_{ij}^{(v)}{f^{(v)}}({x_i}),\quad j = 1, \cdots ,{k_i},i = 1, \cdots ,m,}

\]
where

1
≤

k
i

≤
n

1 \leq {k_i} \leq n

and the

k
i

{k_i}

n-tuples

(

a

i
j

(
0
)

,
⋯
,

a

i
j

(
n
−
1
)

)

(a_{ij}^{(0)}, \cdots ,a_{ij}^{(n - 1)})

are linearly inde pendent. Let

r

i
j

{r_{ij}}

be prescribed real numbers and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U equals left-brace f element-of upper H Superscript n comma normal infinity Baseline colon upper L Subscript i j Baseline f equals r Subscript i j Baseline comma j equals 1 comma midline-horizontal-ellipsis comma k Subscript i Baseline comma i equals 1 comma midline-horizontal-ellipsis comma m right-brace">

U
=
{
f
∈

H

n
,
∞

:

L

i
j

f
=

r

i
j

,
j
=
1
,
⋯
,

k
i

,
i
=
1
,
⋯
,
m
}

U = \{ f \in {H^{n,\infty }}:{L_{ij}}f = {r_{ij}},j = 1, \cdots ,{k_i},i = 1, \cdots ,m\}

. In this paper we consider the extremal problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel double-vertical-bar upper L s double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript Baseline equals alpha equals inf left-brace right-brace colon double-vertical-bar double-vertical-bar times times LfL infinity colon element-of element-of fU period EndLayout">

(

∗

)

‖

L
s

‖

L
∞

=
α
=
inf
{

‖

L
f

‖

L
∞

:
f
∈
U
}
.

\begin{equation}\tag {$\ast $}{\left \| {Ls} \right \|_{{L^\infty }}} = \alpha = \inf \{ {\left \| {Lf} \right \|_{{L^\infty }}}:f \in U\} .\end{equation}

We show that there are, in general, many solutions to

(
∗
)

( \ast )

but that there is, under certain consistency assumptions on L and the

L

i
j

{L_{ij}}

, a fundamental (or core) interval of the form

(

x
i

,

x

i
+

n
0

)

({x_i},{x_{i + {n_0}}})

on which all solutions to

(
∗
)

( \ast )

agree;

n
0

{n_0}

is determined by the

k
i

{k_i}

and satisfies

n
0

≥
1

{n_0} \geq 1

. Further, if s is any solution to

(
∗
)

( \ast )

then on

(

x
i

,

x

i
+

n
0

)
,

|

L
s

|

=
α

({x_i},{x_{i + {n_0}}}),|Ls| = \alpha

a.e. Further, we show that there is a uniquely determined solution

s
∗

{s_ \ast }

to

(
∗
)

( \ast )

, found by minimizing

‖

L
f

‖

L
∞

{\left \| {Lf} \right \|_{{L^\infty }}}

over all subintervals

(

x
j

,

x

j
+
1

)
,
j
=
1
,
⋯
,
m
−
1

({x_j},{x_{j + 1}}),j = 1, \cdots ,m - 1

, with the property that

|

L

s
∗

|

|L{s_ \ast }|

is constant on each subinterval

(

x
j

,

x

j
+
1

)

({x_j},{x_{j + 1}})

and

L

s
∗

L{s_ \ast }

is a step function with at most

n
−
1

n - 1

discontinuities on

(

x
j

,

x

j
+
1

)

({x_j},{x_{j + 1}})

. When

L
=

D
n

,

s
∗

L = {D^n},{s_ \ast }

is a piecewise perfect spline. Examples show that the results are essentially best possible.