, whose elements are defined by
is called a difference matrix, where α and β are real numbers.
Theorem 2 The difference matrix is its own inverse.
. For any double sequence
be a given sequence and
be a diagonal matrix whose only non-zero entries are
. The transformation matrix
is called a double E-J generalized Hausdorff matrix corresponding to the sequence .
Theorem 3 A matrix is a double E
-J generalized Hausdorff matrix corresponding to the sequence if and only if its elements have the form
be a double E-J Hausdorff matrix. Applying this to a double sequence
For double E-J Hausdorff matrices, the necessary and sufficient condition for
to be conservative is the existence of a function
Theorem 4 Given a function
, a bounded variation in the unit square
, the corresponding double E
-J Hausdorff transformation
, of a sequence
, may be defined by
Theorem 5 Let be a conservative double E-J Hausdorff matrix. Then , .
As tools to prove our result, we need to the following lemmas.
Lemma 1 
The following lemma is a double version of .
Lemma 2 For
Proof of Theorem 5
be the double E-J transform of a double sequence
We will demonstrate that
Due to this (1) is equivalent to
It follows from the Hölder inequality that
From Lemma 2
and from Lemma 1
From Lemma of [8
, respectively, then
A similar consequence can be proved for
, see [6
is a moment sequence,
from Theorem 4. In view of (3) we can deduce that
Using Minkowski’s inequality we get
Therefore the proof of Theorem 5 is complete. □
Specially, if we take and in Theorem 5, we get Theorem 1 as a corollary.
The following is an example of a double E-J Hausdorff matrix.
A doubly infinite Cesàro matrix
is a doubly infinite Hausdorff matrix with entries
We use the following to denote the corresponding E-J generalizations of the .
has moment sequence
For the special case
is a double E-J Hausdorff matrix.