Integral operator

An integral operator maps a function $f$ to a function of the formA bounded continuous kernel on a compact domain defines a bounded operator in the uniform norm.

An integral kernel is a function or distribution specifying an integral operator through $(Tf)(x)=\int K(x,y)f(y)dy$. A Green function is an integral kernel that inverts a differential operator under specified boundary conditions.

Suppose an integral kernel reproduces under convolution and has integrable diagonal with integral $r$, and the displayed single-variable integrals exist. In the determinant $D_n=\det [K(x_i,x_j)]$, permutations fixing $n$ contribute $r{D}_{n-1}$ after integration. For every permutation on $n-1$ letters there are $n-1$ ways to insert $n$ into a cycle; convolution contracts the two adjacent factors, and insertion reverses the permutation sign. These terms contribute $-(n-1)D_{n-1}$. This proves the formula without a symmetry assumption on the kernel.

An orthonormal set ${\varphi }_0,\dots ,{\varphi }_{m-1}$ in an $L^2$ space defines the integral kernel of the orthogonal projection onto its span by this sum. It satisfies $\int K(x,y)K(y,z)d\mu (y)=K(x,z)$ and $\int K(x,x)d\mu (x)=m$. Its evaluation matrices are positive semidefinite Gram matrices.

For real monic orthogonal polynomials $p_j$ with squared weighted norms $h_j$ under a positive weight $w$, the functions $p_j\sqrt{w/h_j}$ form an orthonormal set for the unweighted reference measure. Their finite-rank projection kernel turns the squared weighted Vandermonde determinant into a kernel determinant. Normalization on the full labelled configuration space is $1/m!$.

For $w(x)=x^a{e}^{-x}$ on $[0,\infty )$, $a>-1$, the monic Generalized Laguerre polynomials have squared norms $h_j=j!\Gamma (j+a+1)$. The corresponding orthogonal polynomial projection kernel is $(xy{)}^{a/2}{e}^{-(x+y)/2}{\sum }_{j=0}^{m-1}j!L_j^{(a)}(x)L_j^{(a)}(y)/\Gamma (j+a+1)$ for nonnegative $x,y$, and zero otherwise.

One form of the Abel transform isFor sufficiently regular $f$, it is inverted byEndpoint atoms must be handled separately as measures rather than ordinary densities.

The Volterra integration operator on $L^2([0,1])$ is compact and injective, but its inverse is differentiation and is unbounded. Its singular values $2/((2n-1)\pi )$ tend to zero, making recovery of a function from noisy indefinite-integral data an ill-posed inverse problem.

On $(0,1/2)$ use the forward difference and on $(1/2,1)$ the backward difference, with $0<h<1/2$. This defines a bounded linear operator on $L^2(0,1)$ and $\Vert D_h\Vert \le \sqrt{6}/h$. For $f$ in the range of the Volterra integration operator, these differences are local averages of $f^{\prime }$ and converge to $f^{\prime }$ in $L^2$ by continuity of translations. Hence $D_h$ gives a linear regularization of differentiation. No pointwise evaluation of an arbitrary $L^2$ equivalence class is needed: translated functions are defined almost everywhere.

For $f\in C^2[0,1]$ and $\Vert f^{\delta }-f{\Vert }_2\le \delta$, two translated half-interval integrals have overlap multiplicity at most two. The inequality $|a-b{|}^2\le 2|a{|}^2+2|b{|}^2$ gives $\Vert D_h\Vert \le \sqrt{6}/h$. Averaging $f^{\prime }$ over the relevant interval and applying $|f^{\prime }(x+t)-f^{\prime }(x)|\le \Vert f^{\prime \prime }{\Vert }_{\infty }|t|$ gives the second term. For this to estimate a Moore–Penrose inverse of an operator, also require $f(0)=0$; otherwise the derivative is still approximated but $K^{†}f$ is not defined.

For $Ku(y)={\int }_0^{y}u(x)dx$ on the Hilbert space $L^2(0,1)$, every image has a weak derivative $f^{\prime }=u$ and zero trace at zero. Conversely every such Sobolev space element is this integral of its weak derivative. The range is dense because it contains smooth functions supported in $(0,1)$, but it is not closed: a step function can be approximated in $L^2$ by continuous ramps and is not itself in $H^1$. The Moore–Penrose inverse of an operator has this range as its domain and differentiates there.

Mercer's theorem says that a continuous symmetric positive-semidefinite kernel on a compact domain has an orthonormal eigenfunction expansionwith nonnegative eigenvalues and absolute uniform convergence under its standard hypotheses.