Abstract
The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abel summation formulas.
Keywords
- Euler summation
- Abel summation
- bounded variation functions
- special bounded variation functions
- Radon measure
1. Introduction
Abel and the Euler-Maclaurin summation formulas are standard tool in number theory (see e.g. [1, 2]).
The space of
The purpose of this paper is to show that this class of functions (and some subclasses introduced here of function of a single real variable) is the natural settings for (an extended version of) the Euler-Maclaurin formula.
Let us describe now what we prove in this paper.
In Section 2 we obtain some “integration by parts”-like formulas for functions of bounded variations which imply the various “Abel summation” techniques (Propositions (0.6), (0.7), and the relative examples) and in Section 3 we give some criterion for the absolute summability of some series obtained by sampling the values of a bounded variations function.
The last section contains the proofs of the main result of this paper (Theorem (0.1)) that we will now describe.
We denote by
Given
We denote by
Any real function of bounded variation can be written as a difference of two non decreasing functions. It follows that if
Let
We recall that
We denote by
is in
We define
and for each integer
We denote by
where
Moreover
The main results of this paper is the following theorem.
Theorem 0.1 Let
follows easily from Theorem 0.1.
where each
In this paper we do not need of the existence of such a decomposition.
2. Integration by parts formulas
Our starting point is the following theorem:
Theorem 0.2 Let
Since
Since
and hence one obtains the formulas (18) and (19) taking the limits as
Formula (20) is obtained summing memberwise (18) and (19) and dividing by two. □
Next we prove:
Theorem 0.3 Let
where
Moreover, if the function
The function
Since
But
which combined with (31) yields
Using the definition of
and hence
As in the proof of the previous theorem we have
Since
The Radon measure
From (36) it follows that
which is equivalent to (25).
The proof of (26) is obtained in a similar manner using (19) instead of (18), and (27) is obtained summing memberwise (25) and (26) and dividing by two.
If the function
and (29) follows from, e.g., (25).
□
is not, in general, absolutely convergent. Indeed, set
and
Then the integral
is absolutely convergent, but the series
is convergent but not absolutely convergent.
We also have the following theorem.
Theorem 0.4 Let
where
If the function
Since
Then (52) and (53) yield (47). Formulas (48) and (49) are obtained in a similar manner using respectively Formulas (25) and (27) instead of (26).
If the function
and (51) follows from, e.g., (47).□
Theorem 0.4 generalizes to high order derivatives.
Theorem 0.5 Let
Replacing
Replacing
Summing (59) and (60) we obtain (55).
The proofs of (56) and (57) are similar.
□
We say that a function
The following propositions are easy consequences of Theorem (0.4).
Proposition 0.6 Let
Since
The function
and
Summing memberwise the last two formulas we obtain
as desired.□
Proposition 0.7 Let
But then
and
hence
which is equivalent to (67).□
is a step function in
and
hence, Proposition (0.7) yields
3. Sampling estimates
In this section we give some conditions which ensures the absolute convergence of series of the form
The basic estimate is given in the following lemma.
Lemma 0.8 Let
Then, for any complex function
and set
For each
By Eq. (27)
We also have
which implies
Set
Then
and hence
Moreover we have
Taking modules, and observing that
as required.
Corollary 0.9 Let
□
4. Proof of Theorem 0.1
Inserting
By Corollary 0.9 it follows that
is an absolutely convergent series, and hence Theorem 0.1 follows.
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