We study existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives. We develop a natural framework based on Laplace transform as a correspondence between appropriate L-p and Hardy spaces: this point of view allows us to interpret rigorously operators of the form f (partial derivative(t)) where f is an analytic function such as (the analytic continuation of) the Riemann zeta function. We find the most general solution to the equation
f(partial derivative(t))phi = J(t), t >= 0,
in a convenient class of functions, we define and solve its corresponding initial value problem, and we state conditions under which the solution is of class C-k, k >= 0. More specifically, we prove that if some a priori information is specified, then the initial value problem is well-posed and it can be solved using only a finite number of local initial data. Also, motivated by some intriguing work by Dragovich and Aref ‘eva-Volovich on cosmology, we solve explicitly field equations of the form
zeta(partial derivative(t) + h)phi = J(t), t >= 0,
in which zeta is the Riemann zeta function and h > 1. Finally, we remark that the L-2 case of our general theory allows us to give a precise meaning to the often-used interpretation of f (partial derivative(t)) as an operator defined by a power series in the differential operator partial derivative(t).
