Degenerate kernel

Remarks

Example: $K(x, s) = \cos(x - s) = \cos x \cos s + \sin x \sin s$, with $n = 2$, $a_1 = b_1 = \cos$ and $a_2 = b_2 = \sin$.

Non-example: $K(x, s) = e^{x s}$ on $[0, 1]$. If $e^{x s} = \sum_{i=1}^{n} a_i(x) b_i(s)$, then each function $x \mapsto e^{x s}$ would lie in the span of $a_1, \ldots, a_n$. But the functions $e^{s_1 x}, \ldots, e^{s_{n+1} x}$ for distinct $s_j$ are linearly independent, which gives $n + 1$ independent functions in a space of dimension at most $n$.

The names separable kernel and kernel of finite rank are also used. One usually asks that $a_i$ and $b_i$ be continuous, or square-integrable, and one may take $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ linearly independent. For $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$ with a degenerate $K$, every solution has the form $\varphi = f + \lambda \sum_i c_i a_i$ with $c_i = \int_a^b b_i \varphi \, ds$, and the $c_i$ solve an $n \times n$ linear system.