Given n points and n functions, find a linear combination of n functions that fits all n points(details below)?

Question

Fit f(x) = Σ[k=1, n] a_k*g_k(x) to n points, where
①g_1, ..., g_n are linearly independent real functions given.
②all points given are in the domain of all g_1, ..., g_n.
③a_1, ..., a_n are real numbers you pick.

Intuitively, I can set up n simultaneous equations in n unknowns. But is it always the case?

ecapS trebliH · Accepted Answer

I assume that the given functions are defined on some interval J, and are linearly
 
independent as functions on J.

In other words, if c_1 g_1(x) + ...  + c_n g_n(x) = 0

for *every* x in J, then all the coefficients c_k = 0.

Also n points p_1, ..,p_n are given.

It is quite possible that g_k(p_r) = 0 for all k & r.

Trivial example with n = 1:

Given function g(x) = x

Given point p = 0.
--------------
What is needed is that the n functions g_k are linearly independent

functions restricted to the given set S of n points {p_k : 1=<k =< n}

Proof is trivial: the set F(S) of real functions on S is n dimensional,

and the restrictions of the n g's to S are linearly independent, so are

a basis of F(S). That is exactly what is required.

---
Major example: the polynomials 1, x, x^2, ..., x^(n-1).  There is a standard

formula for finding a polynomial P of degree (n-1) with P(x_k) = a_k; it's

a sum of terms a_k Q_k(x)  where Q_k(x_k) = 1 and Q_k(x_j) = 0 otherwise.

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Given n points and n functions, find a linear combination of n functions that fits all n points(details below)?

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