Cantor sets and integral-functional equations
In an earlier paper [ibid. 17, No. 1, 159-181 (1998; Zbl 0897.45004)] the authors have shown that the Fredholm integral equation \[ \phi(t) = (L\phi)(t) = b \int_{at - a + 1}^{at} \phi(\tau) d\tau, \tag{\(*\)} \] where \(a > 1\) and \(b = a/(a-1)\), has a compactly supported solution, which is unique up to a constant scaling factor. Here they primarily investigate the case \(a \geq 2\) closer. In this case the solution is piecewise polynomial. It can be computed recursively, and the authors give explicit formulas for these polynomials. Moreover, for arbitrary \(a > 1\), the iterated kernels of the integral equation \((*)\) are computed, and new spline approximations for the solutions are given.
scientific article; zbMATH DE number 1270174