Linear regression of B-splines with terms inside an integral?

Question

I have encountered a problem that the literature suggests linear regression is able to solve, but I am at a loss.

I have a function $F$ that I want to estimate. This function obeys $N$ equations of the form $$y_i = \int_0^1 F(x) K_i(x) \; \text{d}x \qquad i=1\ldots N$$ where $y_i$ are known data values and $K_i(x)$ are known functions.

I can represent $F$ using cubic B-splines:

$$ F(x) = \sum_{i=-1}^{M+1} a_i B_i (x). $$

I want to approximate $F$ by estimating those coefficients $a$. Hence the first equation can be rewritten as $N$ equations of the following form $$ y_i = a_1 \int_0^1 B_1(x) K_i(x) \;\text{d}x + \ldots + a_M \int_0^1 B_M(x) K_i(x) \;\text{d}x \qquad i=1\ldots N. $$ Now, as I said, the literature says this problem can be solved via linear regression. This would seem to be the case because the coefficients we are estimating are linear in the target. However, what confuses me is that each cubic spline $B$ has to use those coefficients $a$ in order to calculate the result. So how is this linear? What am I doing wrong here?

Answer 1

The $B_i$ are b-spline basis functions. They are completely independent of the $a_i$. So, once the integrals are evaluated, the last set of equations that you wrote is a linear system that you can use to calculate the $a_i$.