Existence of super fast converging Chernoff approximations for Lf = af'' + bf' + cf

Determine whether there exists a practically valuable example of Chernoff approximations that converge faster than 1/n^k for all natural numbers k (super fast convergence) for the second-order differential operator L acting on functions by Lf(x) = a(x) f''(x) + b(x) f'(x) + c(x) f(x), where the coefficient functions a, b, and c are uniformly continuous and bounded.

Background

The paper reviews Dr. Remizov’s work on Chernoff approximations for semigroups generated by differential operators, including constructions that achieve fast convergence for operators of the form Lf = af'' + bf' + cf with uniformly continuous and bounded coefficients.

Within this context, the authors define fast convergence (error decays faster than 1/n) and super fast convergence (error decays faster than 1/n^k for all k ∈ N) and note a method for fast convergence already exists for this operator class. The open problem asks whether a practically valuable example of super fast convergence can be achieved for this operator, indicating a potential significant advancement in the applicability of Chernoff approximations.