Bounded sequence-to-function Hausdorff transformations
Authors: Georgakis Constantine 
Published: July 2010

Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T a right-parenthesis left-parenthesis y right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts left-parenthesis negative y right-parenthesis Superscript n Baseline StartFraction g Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis y right-parenthesis Over n factorial EndFraction a Subscript n comma Baseline y greater-than-or-equal-to 0">

(

T
a

)

(
y
)

=

∑

n
=
0

∞

(

−
y

)

n

g

(
n
)

(
y
)

n
!

a

n
,

y
≥
0

\left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0

\]
be the sequence-to-function Hausdorff transformation generated by the completely monotone function

g
g

or, what is equivalent, the Laplace transform of a finite positive measure

σ
\sigma

on

[
0
,
∞
)

[0,\infty )

. It is shown that for

1
≤
p
≤
∞

1 \leq p \leq \infty

,

T
T

is a bounded transformation of

l
p

{l^p}

with weight

Γ

(

n
+
s
+
1

)

/

n
!

\Gamma \left ( {n + s + 1} \right ) / n!

into

L
p

[
0
,
∞
)

{L^p}[0,\infty )

with weight

y
s

,
s
>
−
1

{y^s},s > - 1

, whose norm
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper T double-vertical-bar equals integral Subscript 0 Superscript normal infinity Baseline t Superscript minus left-parenthesis 1 plus s right-parenthesis slash p Baseline d sigma left-parenthesis t right-parenthesis equals upper C left-parenthesis p comma s right-parenthesis">

‖
T
‖

=

∫
0
∞

t

−

(

1
+
s

)

/

p

d
σ

(
t
)

=
C

(

p
,
s

)

\left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right )

if and only if

C

(

p
,
s

)

>
∞

C\left ( {p,s} \right ) > \infty

, and that for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity comma double-vertical-bar upper T a double-vertical-bar Subscript p comma s Baseline greater-than upper C left-parenthesis p comma s right-parenthesis double-vertical-bar a double-vertical-bar Subscript p comma s Baseline">

1
>
p
>
∞
,

‖

T
a

‖

p
,
s

>
C

(

p
,
s

)

‖
a
‖

p
,
s

1 > p > \infty ,{\left \| {Ta} \right \|_{p,s}} > C\left ( {p,s} \right ){\left \| a \right \|_{p,s}}

unless

a
n

{a_n}

is a null sequence. Furthermore, if

1
>
p
>
r
>
∞
,

0
>
λ
>
1

1 > p > r > \infty ,\,\;0 > \lambda > 1

and

σ
\sigma

is absolutely continuous with derivatives

ψ
\psi

such that the function

ψ
r

(
t
)

=

t

−
1

/

r

ψ

(
t
)

{\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right )

belongs to

L

1

/

λ

[
0
,
∞
)

{L^{1 / \lambda }}[0,\infty )

, then the transformation

(

T
λ

a

)

(
y
)

=

y

1
−
λ

(

T
a

)

(
y
)

\left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right )

is bounded from

l
p

{l^p}

to

L
r

[
0
,
∞
)

{L^r}[0,\infty )

and has norm

‖

T
λ

‖

≤

‖

ψ
r

‖

1

/

λ

\left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }}

. The transformation

T
T

includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on

l
p

{l^p}

which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.