Extending a discrete singular kernel Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:

*

*$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n.$


*$|\phi(n+1)-\phi(n)|\leq \frac{C_2}{n^2}$ for all $n\neq 0$ and $C_2>0$ is independent of $n.$


*$\sum_{-N}^N\phi(n)$ converges as $N\to\infty.$
Denote
$$
R(x):=
\begin{cases}
1-|x| & \text{for }|x|<1\\
0 &\text{otherwise}
\end{cases}.
$$
and consider
$$
K(x)=\sum_{n\in\mathbb Z}\phi(n)R(x-n),
$$ which exists as a function on $\mathbb R.$

*

*Can anyone prove that
$$
\lim_{\epsilon\to \infty}\int\limits_{\frac{1}{\epsilon}<|x|<\epsilon}K(x)\,\mathrm{d}x
$$ exists?

*Also are the following true?
$$
\begin{align}
|K(x)-K(x-y)| &\leq C_3\frac{|y|}{|x|^2} &\text{for }|x|>2|y|>0\\ 
|K(x)| & \leq C_4|x|^{-1}&\text{for }x\neq 0.
\end{align}
$$ Also it is okay to have the above conditions true almost everywhere.

 A: $\newcommand\R{\mathbb R}\newcommand{\Z}{\mathbb{Z}}\newcommand{\ep}{\epsilon}\newcommand{\fl}[1]{\lfloor#1\rfloor}$The answer is yes to each of your two questions.
Let $a_n:=\phi(n)$. Then
\begin{equation*}
    K(x)=\sum_{n\in\Z}a_n R(x-n). 
\end{equation*}
Note that for all $j\in\Z$ we have $K(j)=a_j$ and $K$ linear (or, more exactly, affine) on the interval $[j,j+1]$. Also, $K$ is continuous. So, $K$ is obtained by the linear interpolation of the function $\Z\ni j\mapsto a_j$. In particular, $K$ is bounded.
So, for real $\ep>0$,
\begin{equation*}
    \int_{1/\ep<|x|<\ep}K(x)\,dx=I_\ep+O(1/\ep),
\end{equation*}
where
\begin{equation*}
    I_\ep:=\int_{|x|<\ep}K(x)\,dx
    =\sum_{n\in\Z}a_n J_n,
\end{equation*}
\begin{equation*}
    J_n:=\int_{-\ep}^\ep dx\,R(x-n). 
\end{equation*}
Let now $N:=\lfloor\ep\rfloor$, so that $N\le\ep<N+1$. Then $J_n=\int_\R dx\,R(x-n)=1$ if $|n|\le N-1$ and $J_n=0$ if $|n|\ge N+2$. Also, $0\le J_n\le1$ for all $n\in\Z$. So,
\begin{equation*}
    I_\ep
    =\sum_{|n|\le N-1}a_n +O(|a_N|+|a_{-N}|+|a_{N+1}|+|a_{-N-1}|). 
\end{equation*}
So, $I_\ep$ converges, since $N\to\infty$ (as $\ep\to\infty$), $\sum_{|n|\le N-1}a_n$ converges, and $|a_N|+|a_{-N}|+|a_{N+1}|+|a_{-N-1}|=O(1/N)\to0$. So, $\int_{1/\ep<|x|<\ep}K(x)\,dx$ converges.
This provides the positive answer to your first question.
The answer to your second question is also positive, that is, for some real $C_3$ and $C_4$,
\begin{equation*}
    |K(x)|\le C_3|x|^{-1} \text{ if }x\ne 0 \tag{1}
\end{equation*}
and
\begin{equation*}
    |K(x)-K(x-y)| \le C_4\frac{|y|}{|x|^2}  \text{ if }|x|>2|y|>0. \tag{2}
\end{equation*}
Indeed, since $K$ is obtained by the linear interpolation of the function $\Z\ni j\mapsto a_j$ and the $a_j$'s are bounded, we see that the function $K$ is bounded and Lipschitz, so that without loss of generality $|x|>8$ in (1) and (2).
Now take indeed any real $x$ with $|x|>8$ and any real $y$ as in (2). Let
\begin{equation}
    j:=\fl{|x|},\quad m:=\fl{|x-y|},
\end{equation}
so that $|j|\ge|m|\ge1+|x|/4$ and also $|j+1|\ge|x|/4$.
So, by the linear interpolation observation and the condition $|a_n|\le C_1/|n|$ for $n\ne0$,
\begin{equation*}
    |K(x)|\le|a_j|+|a_{j+1}|\le C_1(|j|^{-1}+|j+1|^{-1})\le8C_1|x|^{-1}, 
\end{equation*}
which verifies (1).
Next, by the linear interpolation observation and the condition $|a_{n+1}-a_n|\le C_2/n^2$ for $n\ne0$, the function $K$ is Lipschitz on $[m,\infty)$ and on $(-\infty,-m]$ with Lipschitz constant $C_2/(m-1)^2\le4C_2/x^2$.
Also, $|x|\ge j\ge m$, $|x-y|\ge m$, and, by the condition $|x|>2|y|>0$ in (2), $x$ and $x-y$ are of the same sign. So, either both $x$ and $x-y$ are in $[m,\infty)$ or they are both in $(-\infty,-m]$.
Therefore and because the function $K$ is Lipschitz on $[m,\infty)$ and on $(-\infty,-m]$ with Lipschitz constant $4C_2/x^2$, (2) follows.
A: Come on, for the first property just use the fact that
$$ \int_{-n}^n K(x) dx = \sum_{j = -n+1}^{n-1} \phi(j) + \tfrac12(\phi(-n)+\phi(n)) $$
has a finite limit as $n \to \infty$, together with convergence of
$$ \biggl| \int_{-a}^a K(x) dx - \int_{-\lfloor a\rfloor}^{\lfloor a\rfloor} K(x) dx \biggr| \leqslant |\phi(-\lfloor a\rfloor-1)| + |\phi(-\lfloor a\rfloor)| + |\phi(\lfloor a\rfloor)| + |\phi(\lfloor a\rfloor+1)| $$
to zero as $a \to \infty$.
The other two properties are equally simple, once you realise that
$$ |\phi(n) - \phi(n - k)| \leqslant \sum_{j = 1}^k |\phi(n - j) - \phi(n - j + 1)| \leqslant \sum_{j = 1}^k \frac{C_2}{|n - j|^2} \leqslant C_2 \frac{k}{(|n| - |k|)^2} . $$
