Meaning of "generalization"? in math

Question

What is the meaning of word "generalization"? Please kindly explain in simple words

For example in wikipedia article of interpolation, in sub heading "polynomial interpolation " a line is written

Polynomial interpolation is a generalization of linear interpolation

What does it means? Polynomial interpolation is a sub-branch of the main branch linear interpolation? Or its opposite?

Another example from a text of signal processing: "FIR filter is a generalization of the idea of a running average "

Answer 1

Theorem $A$ is a generalization of Theorem $B$ if the result in $B$ is a special case of Theorem $A$. For example:

Theorem $A$: Every integer divisible by $2$ and $3$ is also divisible by $6$.

Theorem $B$: $12$ is divisible by $6$.

Theorem $B$ is a special case of Theorem $A$ when we let our integer to be $12$.

Another example:

Theorem $A$: Given $n+1$ points $(x_i,y_i)$ there is a polynomial of degree $n$ which passes through them.

Theorem $B$: Given $2$ points $(x_1,y_1),(x_2,y_2)$ there is a linear polynomial which passes through them.

Theorem $B$ is a special case of Theorem $A$ when $n=1$.

The same applies to definitions as well. Definition C generalizes Definition D if the latter is a special case of the former.