Why polynomial interpolation is considered as better than others?

Question

Why polynomial interpolation is considered as better than others?

In case of interpolation, the function $\phi(x)$ to approximate the unknown function $f(x)$ may be polynomial, exponential, trigonometric sum, Taylor's series, piece-wise polynomial etc. Then why polynomial interpolation is considered as better than others (although I know that there is a justification for approximation by polynomials (Weierstrass's theorem), but it does not true that there is no justification to approximate by exponential, trigonometric etc).

Some where I read, reason for considering the polynomials in the approximation of functions $f(x)$ is that the derivative and indefinite integral of a polynomial are easy to determine and are also polynomials. But if $\phi(x)$ is exponential, trigonometric sum, Taylor's series, piece-wise polynomial etc then also derivative and indefinite integral are determined like polynomials.

Then what are other reason(s) for taking polynomials into consideration?

Answer 1

The other reason is probably that comparing the polynomials to other elementary functions the polynomial are also easier to compute, this is a practical justification of course. Theoretically speaking... i'm not sure of this statement because you can also approximate a function using the trigonometric polynomial. Trigonometric functions aren't easy to compute in practice (in a computer i mean) but of course are important for solve in closed form certain differential equations. I don't know about the exponential, but they have other applications, like in functional analysis. I believe it actually depends from the application rather than only theoretical properties.

I don't know your background of course but if you try to implement from nothing a trigonometric function (in its whole domain with enough accuracy) you would see a lot of issues...