Multiplication with discrete delta function

Question

What would be the outcome of multiplying the delta function of discrete time with another function

e.g.: δ(n-m)•y(n) =?

What would be the outcome of the multiplication between two different deltas?

e.g.: δ(n)•δ(n-m)=?

Answer 1

I'm not sure if you're referring to the Dirac delta function $\delta(x)$ or the discrete delta function $\delta(x)$defined as follows.

$$\delta(x)=\left\{ \begin{array}{cc} 1 & x=0 \\ 0 & \text{True} \\ \end{array} \right.\tag{1}$$

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The Dirac delta function $\delta(x)$ is mostly useful in integrals such as the following.

$$\int\limits_{-\infty}^\infty\delta(n-m)\ y(n)\,dn=y(m),\quad m\in\mathbb{R}\tag{2}$$

$$\int_{-\infty }^\infty\delta(n)\ \delta(n-m)\,dn=\delta(m)\tag{3}$$

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For the the discrete delta function:

$$\delta(n-m)\ y(n)=\left\{ \begin{array}{cc} y(n) & m=n \\ 0 & \text{True} \\ \end{array} \right.\tag{4}$$