How to Construct a Interpolating Function f(n) such that f(1)=a1, f(2)=a2, ..., f(n)=an, and f(x)=0 for all the other integer x?

Question

How to construct a interpolating function f(n) such that f(1)=a1, f(2)=a2, ..., f(n)=an, and f(x)=0 for all the other integer x ? Recently, I learn about the Stirling numbers of the second kind https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind And find it has such an interesting property So I wonder how to construct such function according to any given data? Construct a f(n) satisfied f(1)=a1, f(2)=a2, ..., f(n)=an, and f(x)=0 for all the other integer x

Answer 1

As I understand it, you are requiring a $f\; : \; {\mathbb R} \to {\mathbb R}$ such that $$ f(x) = \left\{ {\begin{array}{*{20}c} {a_k } & {\left| {\;x = k \in \left\{ {1,2, \ldots ,n} \right\}} \right.} \\ 0 & {\left| {\;x \in {\mathbb Z}\backslash \left\{ {1,2, \ldots ,n} \right\}} \right.} \\ \end{array}} \right. $$ If my understanding is correct, then you can take the product of a Boxcar function of unitary value times the polynomial interpolating $a_1 , a_2, \cdots , a_n$ .

If you are looking for a continuous function, it shall have an infinite number of zeros, so cannot be a polynomial of finite order.
In this case the boxcar can be replaced by the sum of $n$ sinc functions centered at $x=1,2, \ldots, n$.