Approximation of a kernel by degenerate kernels

Remarks

Part 1 follows from the Stone–Weierstrass theorem: polynomials in $x$ and $y$ are dense in the continuous functions on the square. Each monomial $x^p y^q$ is a product of a function of $x$ and a function of $y$. For part 2, let $(e_i)$ be an orthonormal basis of $L^2[a, b]$. The products $e_i(x) \overline{e_j(y)}$ form an orthonormal basis of $L^2([a, b]^2)$. The partial sums of the expansion of $K$ in this basis are degenerate kernels converging to $K$.

Example: $K(x, y) = e^{xy}$ on $[0, 1]^2$ is approximated by $\sum_{k=0}^{N} x^k y^k / k!$, with error at most $\sum_{k > N} 1/k!$.

The norm of an integral operator is at most the $L^2$ norm of its kernel, and at most $b - a$ times its sup norm. So the result says that the integral operator of $K$ is a norm limit of finite-rank operators. This is the main step in the proof that such operators are compact, and in the reduction of Fredholm's theorem to linear algebra.

Going deeper

For a student

The approximation is used together with a perturbation argument. Write $K = K_\varepsilon + (K - K_\varepsilon)$; the small part is inverted by a Neumann series, and the degenerate part leads to a finite linear system. This yields the Fredholm alternative for square-integrable kernels, and the compactness of their integral operators.