 Singular integral operator

A singular integral operator has a kernel whose ordinary integral can fail at points on the integration domain. A specified limiting prescription, such as a Cauchy principal value, can still define the operator by cancellation. For the Hilbert transform kernel $1/(t-x)$, subtracting $f(x)$ from a locally Hölder-continuous density leaves $[f(t)-f(x)]/(t-x)$ locally integrable; the constant part is interpreted by symmetric deletion. The limiting prescription and the class of allowed densities are part of the operator's definition, not optional details.