**4. Proof of Theorem 0.1**

Inserting *Bm*ð Þ *x* instead of *gm*ð Þ *x* in formula (57) of Theorem 0.5 we easily obtain

$$\begin{split} \sum\_{n \in \mathbb{Z}} \frac{f(n^{+}) + f(n^{-})}{2} &= \int\_{\mathbb{R}} f(\mathbf{x}) d\mathbf{x} + \sum\_{\mathbf{x}} \sum\_{k=1}^{m} \frac{(-1)^{k-1}}{k!} B\_{k}(\mathbf{x}) \delta f^{(k-1)}(\mathbf{x}) \\ &+ \frac{(-1)^{m-1}}{m!} \int\_{\mathbb{R}} B\_{m}(\mathbf{x}) f^{(m)}(\mathbf{x}) d\mathbf{x}. \end{split} \tag{91}$$

By Corollary 0.9 it follows that

$$\sum\_{n \in \mathbb{Z}} \prime \frac{f(n^+) + f(n^-)}{2} = \sum\_{n \in \mathbb{Z}} \frac{f(n^+) + f(n^-)}{2} \tag{92}$$

is an absolutely convergent series, and hence Theorem 0.1 follows.

*Abel and Euler Summation Formulas for* SBV *(*R*) Functions DOI: http://dx.doi.org/10.5772/intechopen.100373*
