		\begin{align*}
		&\frac{1}{\tilde{T}h} \sum_{k=1}^{\tilde{T}} K_h(x,x_k)g(x_k)=g(x) + \int_{-\infty}^\infty h^{-1}\mathcal{K}_h(x,t) {(x-t)^{q} \frac{g^{(q)}(\xi_{x,t})}{q!}}\, \mbox{d}t\ +\mathcal{O}(\tilde{T}^{-\beta})\\
		&= g(x) +  \int_{-\infty}^\infty h^{-1}\mathcal{K}_h(x,t) {(x-t)^{q} \frac{g^{(q)}(\xi_{x,t}) - g^{(q)}(x)}{q!}}\, \mbox{d}t  + \mathcal{O}(\tilde{T}^{-\beta})\\
		&= g(x) + \sum_{l=-\infty}^\infty \int_{x+(l-1)h}^{x+lh} \mathcal{K}_h(x, t)(x-t) ^{q} \frac{g^{(q)}(\xi_{x,t})-g^{(q)}(x)}{hq!}\, \mbox{d}t+ \mathcal{O}(\tilde{T}^{-\beta}),
		\end{align*}
