Are Polynomial Interpolation and Taylor Series expansion somehow related?
I'm a high school student and in our Statistics class today, we were given a very brief introduction about interpolation. Now we were told interpolation is used to approximate a function by some given values at different points on $x$-axis. I know this is very introductory and there's a lot more to interpolation that I don't know, but I can't help but think if I can somehow connect interpolation with sine function to get a polynomial expansion of sine function. I'm aware of the Taylor Series expansion of sine function, which represents sine as an infinite polynomial. My question is that, is interpolation used there? We know values of sine at different points, can we approximate it's polynomial expansion from interpolation?
Best Answer
great questions! When you express the sine function as an infinite summation of powers of x you are not "interpolating" anything. Interpolation in math is typically the act of taking some points, and connecting them. More formally...
Interpolation is a method for finding a function that fits a set of data points. In other words, if you have a set of points $(x_1, y_1), (x_2, y_2), … (x_n, y_n),$ interpolation can be used to find a function $f(x)$ that passes through all of those points. This function is called an interpolating function.
The Taylor series expansion of a function is a way of representing that function as an infinite sum of terms, each of which is a polynomial. This is different from interpolation, which is used to find a single function that fits a set of data points.
It is possible to use interpolation to approximate the Taylor series expansion of a function, but it is not the only way to do so. In general, the Taylor series of a function can be found by taking the derivative of the function at a point and using that information to find the coefficients of the polynomial terms in the series. This method does not require interpolation.
In the case of the sine function, the Taylor series is:
$\sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...$
This series can be derived using calculus, without needing to use interpolation.
We can use interpolation to approximate not only $sin(x)$, but also each of the terms that make up it's Taylor expansion and all linearly combinations. That is many different functions. Furthermore, we have infinite freedom as to how we interpolate these functions. For example, we can choose to interpolate with 60 points, or 120 points, and we can choose to linearly space these points, or perhaps space them as the roots of the Chebyshev polynomials of the first kind.
Some good problems to try solving are:
Write down a polynomial that interpolates the points $(x_1, y_1), (x_2, y_2), … (x_n, y_n)$.
Explore the question of existence and uniqueness for interpolating polynomials when we have n points: $(x_1, y_1), (x_2, y_2), … (x_n, y_n)$.