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.
- Euler summation
- Abel summation
- bounded variation functions
- special bounded variation functions
- Radon measure
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 (resp. , and respectively the space of continuously differentiable functions (resp. -times differentiables functions on the closed interval ), the space of Lebesgue (absolutely) integrable functions and the space of essentially bounded Borel functions on .
Given and we set
Any real function of bounded variation can be written as a difference of two non decreasing functions. It follows that if then , and exist for each and the set is an arbitrary at most countable subset of . Moreover, the derivative exists for almost all and .
Let . We denote by the unique Radon measure on such that for each open interval
We recall that is
We denote by the space of all special functions of bounded variation. We also say that (resp. ) if for each , with the function
is in (resp. ).
We define inductively setting
and for each integer
We denote by and , respectively the Bernoulli numbers and the Bernoulli functions. Let us recall that
where stands for the greatest integer less than or equal to and , are the unique continuous functions such that
Moreover for and for .
The main results of this paper is the following theorem.
Theorem 0.1 Let , and suppose . Then
follows easily from Theorem 0.1.
where each is a piece-wise constant function, is an absolutely continuous function and is a singular function, that is is continuous and for almost all . Then is special if, and only if, and in this case, for each bounded Borel function ,
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 two complex function. Assume that and . Then
Since then necessarily
Since then and are bounded and we also have
Next we prove:
Theorem 0.3 Let two complex function. Assume that and and suppose that . Then
Moreover, if the function also is continuous then
The function is in . Hence, formula (18) yields
Since we have
But for almost all and hence
which combined with (31) yields
Using the definition of we have
As in the proof of the previous theorem we have
Since and then and hence
The Radon measure is bounded and the functions are equibounded with respect to and ; by the Lebesgue dominated convergence we have
From (36) it follows that
which is equivalent to (25).
If the function is continuous then for each ,
is not, in general, absolutely convergent. Indeed, set
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 two complex function. Assume that and and suppose that . Then
If the function also is continuous then
Since , using the fact that for almost all , we obtain
If the function is continuous then for each ,
Theorem 0.4 generalizes to high order derivatives.
Theorem 0.5 Let two complex function. Let be a positive integer. Assume that with and with . Then
Replacing with , with and changing the sign we obtain
Replacing with in (47) we obtain
We say that a function is a
The following propositions are easy consequences of Theorem (0.4).
Proposition 0.6 Let be a bounded closed interval and let be an absolutely continuous function on the closed interval . Let be a step function. Then
Since is a step function then for almost all and hence it follows that
The function by construction has compact support, and hence, as , we have
Summing memberwise the last two formulas we obtain
Proposition 0.7 Let be two step function. Let be a bounded closed interval. Then
which is equivalent to (67).□
is a step function in . If then Proposition (0.6) yields
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 where is a function absolutely integrable of bounded variation and is a countable subset of .
The basic estimate is given in the following lemma.
Lemma 0.8 Let be an open subset and let be a finite subset of . Assume that there exist such that
Then, for any complex functionwe have
For each we have
By Eq. (27)
We also have
Moreover we have if and hence
Taking modules, and observing that for each , we obtain
Corollary 0.9 Let and let be a countable subset. If there exists a real constant such that for each pair of distinct we have then