I am trying to complete a computer program written by someone who is unfortunately now deceased. Sparing the details, the program computes a kernel $K$ intended to be used in the following integral equation:
\begin{equation} a = \int_0^1 K(x)f(x) \,\text{d}x. \end{equation}
where $a$ is a real-valued positive number and $f$ is a real-valued non-negative function. The functional form of $K$ is long and probably not worth putting here, but in summary, its functional form is derived from physics equations, and the program solves for $K$ numerically using the results of a simulation. [This form I have presented here is actually a slight simplification, as in the real formulation there is actually a sum of two kernels operating on two variables, but hopefully it is not an oversimplification.]
The program furthermore computes and removes the complementary function $T$, by which I mean the function such that $$\int_0^1 T(x) f(x) \, \text{d}x = 0.$$
The complementary function is removed in the following manner:
\begin{align*} \text{let} \quad \alpha &= \frac{\int_{0}^1 K T \, \text{d}x} { \int_0^1 T^2 \, \text{d}x} \\ \text{then} \quad K &:= K - \alpha T. \end{align*}
However, this manner of removing the complementary function is not documented, nor is it mentioned in any of the dozens of papers that make use of this program, other than the sentence that "$K$ is arbitrary up to a multiple of $T$ and hence $T$ can be removed from $K$."
Question 1: Is there a reference someone can recommend for understanding why the complementary function is removed in this way?
The program goes on to obtain a new kernel, let us call it $K_2$, which is the kernel for a different variable, $f_2$:
$$a = \int_0^1 K(x)f(x) \,\text{d}x = \int_0^1 K_2(x) f_2(x)\,\text{d}x.$$
This new kernel $K_2$ is found by solving a boundary value problem involving $K$. [If it is useful, I can post the details of this boundary value problem, but it is again somewhat involved. If it happens to be useful information, $f_2 = g/f_1$ where $g$ is another real-valued non-negative function.]
However, no complementary function $T_2$ is removed from $K_2$.
I have tried to find $T_2$ by solving the boundary value problem where I have substituted $T$ in the place of $K$, but this crashes my solver, and furthermore, I do not know if this is rigorously the right thing to do to find it.
Question 2: How can I find the complementary function $T_2$ such that $\int_0^1 T_2 f_2\,\text{d}x = 0$?