Polynomial interpolation of operators

The numerical approximation of nonlinear functionals and functional differential equations

“…Differently from interpolation methods in spaces of finite dimension (e.g., d-dimensional Euclidean spaces), interpolation here is in a space of functions, i.e., the interpolation nodes θ k (x) are functions in a Hilbert or a Banach space. Over the years, the problem of constructing a functional interpolant through suitable nodes in Hilbert or Banach spaces has been studied by several authors and convergence results were established in rather general cases [134,103,183,178,99,4,106,104,179,216]. Before discussing functional interpolation in detail, let us provide some geometric intuition on what functional interpolation is and what kind of representations we should expect.…”

“…A nonlinear functional F is in fact a particular type of nonlinear operator from a space of functions D(F ) (the domain of the functional F ) into a vector space, e.g., R or C. Thus, the problem of approsimating nonlinear functionals is basically the same as approximating nonlinear operators. This topic has been studied extensively by different scientific communities (see, e.g., [216,199,155,205,70,183,17,134,104]) for obvious reasons. What does it mean to approximate a nonlinear functional?…”

“…Porter [178] applied Prenter's theorems to causal systems, while Bertuzzi, Gandolfi and Germani [16,17] extended Prenter's results to causal approximation of input-output maps in Hilbert spaces. Generalizations to Banach spaces can be found in Chapter 3 of [216] (see also [134] and the references therein). The functional derivatives of Prenter's polynomial functionals can be obtained by computing the functional derivatives of (150).…”

“…We have seen that any continuous functional in a Hilbert or a Banach space can be approximated uniformly in terms of polynomial functionals [181,17,94,134,92,216]. Such polynomial functionals may be built based on an interpolation process (see Section 3.2.3 and Section 3.2.4).…”

“…In this report we will present various techniques for nonlinear functional approximation, ranging from polynomial functional series expansions, to expansions based on stochastic processes, and functional tensor methods. Within the context of polynomial functional series expansions we will discuss in particular Lagrange interpolation in Hilbert and Banach spaces [134,183,181,17,4,28], where the interpolation "nodes" are functions in a suitable function space. We will also discuss series expansions based on functional tensor methods [78].…”