Classical Theorems on Singular Integrals

By analogy, we first investigate the conditions that the kernel \(\varphi \) must satisfy for the integral \(I_n\) to exist and converge to f(x) as n increases, with f(t) being a continuous function at \(t = x\) and belonging to one of the five families considered, in (0, l), in the preceding chapter. The reasoning in this chapter immediately leads to the sought-after conditions, which I will state using the term interior in the strict sense. Specifically, a point x is considered interior to (0, l) if \(0 < x < l\) , excluding the endpoints. Similarly, (a, b) is interior (or completely interior) to (0, l) if \(0 < a < b < l\) .