Asymptotic solutions of integral convolution equations on large or small intervals under weak kernel assumptions
Room Coriolis, Galois building – June 24, 2019 at 10.30 AM.
One-dimensional convolution integral equations with real-valued regular symmetric kernels naturally arise in a number of inverse problems for elliptic PDEs when field values on an one-dimensional set are known whereas the source term is not. In certain geometrical configurations this can be rephrased as an issue of inversion of a compact self-adjoint integral operator, a problem that can be solved (with automatic regularization) by constructing an approximation of the resolvent of this operator, given its approximate eigendecomposition.
Motivated by this application and number of other contexts where convolution integral operators play a crucial role (such as approximation and probability theory, signal processing, physical problems in radiation transfer, neutron transport, quantum gases statistics, geological prospecting, etc), we consider a generic eigenvalue problem for one-dimensional convolution integral operator on an interval where the kernel is real-valued even C1-smooth function which (in case of large interval) is absolutely integrable on the real line.
We show how this spectral problem can be solved by two different asymptotic techniques that take advantage of the size of the interval. In case of small interval, this is done by approximation with an integral operator for which there exists a commuting differential operator thereby reducing the problem to a boundary-value problem for second-order ODE, and often giving the solution in terms of explicitly available special functions such as prolate spheroidal harmonics. In case of large interval, the solution hinges on solvability, by Riemann-Hilbert approach, of an approximate auxiliary integro-differential half-line equation of Wiener-Hopf type, and culminates in simple characteristic equations for eigenvalues, and, with such an approximation to eigenvalues, approximate eigenfunctions are given in an explicit form.
Besides the crude periodic approximation of Grenader-Szego, since 1960s, large-interval spectral results were available only for integral operators with kernels of a rapid (typically exponential) decay at infinity or those whose symbols are rational functions. We assume the symbol of the kernel, on the real line, to be continuous and, for the sake of simplicity, strictly monotonically decreasing with distance from the origin. Contrary to other approaches, the proposed method thus relies solely on behavior of the kernel’s symbol on the real line rather than the entire complex plane which makes it a powerful tool to constructively deal with a wide range of integral operators. We note that, unlike finite-rank approximation of a compact operator, the auxiliary problems arising in both small- and large-interval cases admit infinitely many solutions (eigenfunctions) and hence structurally better represent the original integral operator. The present talk covers an extension and significant simplification of the previous author’s result on Love/Lieb-Liniger/Gaudin equation [to appear in Integral Methods in Science & Engineering, Springer (2019)].