Which Machine or deep learning algorithm is appropriate of this issue?
Your question can be interpreted in many ways. First, convolution with a number is not a common operation. Convolution is defined by:

$$(f*g)=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau$$

Assuming a constant signal $g(t)=a\in \mathbb{R}$ this is reduced to:
$$(f*g)=\int_{-\infty}^{\infty}af(\tau)d\tau=a\int_{-\infty}^{\infty}f(\tau)d\tau$$ 

I am assuming, therefore, that you meant multiplication and not convolution. Correct me otherwise and I will edit the answer accordingly. 

Secondly, assume for your example the given parameters:

$$Y=[1,2,3,4],J=\left[
 \begin{matrix}
  1 &amp; 2 &amp; 3 &amp; 4\\
  1 &amp; 2 &amp; 3 &amp; 4\\
  0 &amp; 0 &amp; 0 &amp; 0\\
  0 &amp; 0 &amp; 0 &amp; 0
 \end{matrix}
\right], X=\left[\begin{matrix}4 \\ 4\\0\\0\end{matrix}\right]$$

It is easy to see that a couple of use-cases could give us this result. For example, it could be that the chosen $y$ was $y_4=1$ and the chosen column was the most right column $j_3$. On the other hand, the combinations $\{y_1,j_1\},\{y_3,j_0\}$ are also plausible. I have purposedly gave a very bad, linearly dependent, matrix $J$ to illustrate that the choice of $J$ in the design process or formulation of the problem is of major interest and is fundamental for answering you question. The following example illustrates the complete opposite:

$$Y=[1,2,3,4],J=\left[
 \begin{matrix}
  1 &amp; 0 &amp; 0 &amp; 0\\
  0 &amp; 1 &amp; 0 &amp; 0\\
  0 &amp; 0 &amp; 1 &amp; 0\\
  0 &amp; 0 &amp; 0 &amp; 1
 \end{matrix}
\right], X=\left[\begin{matrix}x_0 \\ x_1\\x_2\\x_3\end{matrix}\right]$$

Obviously, $x_0=y*j_0$,$x_1=y*j_1$, and so on. Here, it is easy to deduce $y$ and only a small number of $X$ are even possible. 

Generally, if we call $J$ a dictionary, you are looking at something called Orthogonal Matching Pursuit (OMP) problem, which is not a machine or deep learning problem. Otherwise, you can try and use a deep learning model to try and learn the dictionary, though this would be unrecommended as such a solution is not compatible with such a problem in my opinion. 

Hope this answers your question. If not, or I misinterpreted your question, feel free to comment and I will try to supply more information if I can.