## 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 * special functions of bounded variation* (

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 * special* if for any bounded Borel function

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

** Remark.** The sum “

** Remark.** Let

follows easily from Theorem 0.1.

** Remark.** Any

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

** Proof:** 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

** Proof:** Given

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).

□

** Example.** This example shows that in the hypoteses of Theorem (0.3) the series

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

** Proof:** Let

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

** Proof:** We prove first the formula (55). The proof is by induction on

Replacing

Replacing

Summing (59) and (60) we obtain (55).

The proofs of (56) and (57) are similar.

□

We say that a function * step function* if

The following propositions are easy consequences of Theorem (0.4).

Proposition 0.6 Let

** Proof:** First we extend the functions

Since

The function

and

Summing memberwise the last two formulas we obtain

as desired.□

Proposition 0.7 Let

** Proof:** Set both the functions

But then

and

hence

which is equivalent to (67).□

** Example 1.** (Abel summation I) Let

is a step function in

** Example 2.** (Abel summation II) Let

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

** Proof:** Let define

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

** Proof:** It suffices to choose

□

## 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.

## References

- 1.
Graham Everest and Thomas Ward. An Introduction to Number Theory . Springer-Verlag, 2005. - 2.
Henri Cohen. Number Theory Volume II: Analytic and Modern Tools , volume 240 ofGraduate Texts in Mathematics . Springer-Verlag, 2007. - 3.
Ennio De Giorgi and Luigi Ambrosio. New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) , 82(2):199–210 (1989), 1988. - 4.
L. Ambrosio, M. Miranda, Jr., and D. Pallara. Special functions of bounded variation in doubling metric measure spaces. In Calculus of variations: topics from the mathematical heritage of E. De Giorgi , volume 14 ofQuad. Mat. , pages 1–45. Dept. Math., Seconda Univ. Napoli, Caserta, 2004. - 5.
Stanislaw Lojasiewicz. An Introduction to the Theory of Real Functions . Wiley, 1988. - 6.
Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Function of Bounded Variation and Free Discontinuity Problems . Oxford Mathematical Monographs. Oxford University Press, 2000. - 7.
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics . Addison-Wesley Publishing Company, 1989.