I Proving Newton's forward difference interpolation formula

AI Thread Summary
The discussion focuses on proving Newton's forward difference interpolation formula, expressed as a polynomial involving coefficients a_n derived from finite differences. The formula starts with the base case where y_0(x_0) equals a_0 and extends to higher-order terms using finite differences. The key point is that the coefficients a_n can be calculated as a_n = (Δ^n y_0) / (h^n n!), where h represents the step size. The conversation also touches on using induction or a triangular matrix approach to derive these coefficients systematically. Overall, the goal is to establish a rigorous proof of the interpolation formula.
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How to prove newtons forward difference interpolation formula using induction?
Say, $$y_n (x) = a_0 + a_1(x -x_0) + a_2(x-x_1)(x - x_0) + ... +a_n(x-x_0)(x-x_1)...(x-x_{n-1})$$
Now, $$y_0(x_0) = a_0$$
$$y_1(x_1) = a_0 + a_1(x_1 - x_0)$$
or, $$a_1 = \frac{\Delta y_0}{h}$$
Here, $$h = \frac{x_i - x_0}{i}$$
Similarly, $$a_n = \frac{(\Delta)^n y_0}{h^n n!}$$

Next substituting the values of a, we get the Newton's Forward Interpolation Formula.

It is not difficult to see that ##a_n = \frac{(\Delta)^n y_0}{h^n n!}##. But how do I prove this by induction method? Or any other rigorous way?

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