An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function @math on the reals, and for any positive continuous function @math , it has a @math solution with @math for all @math . Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE has a unique solution, even with a finite number of conditions of the form @math . The question whether one can require the solution that approximates @math to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a polynomial ODE such that for any @math and @math there exists some initial condition that yields a solution that is @math -close to @math at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.
All the results mentioned so far are concerned with approximations of continuous functions over the whole real line. Approximating functions over a compact domain seems to be a different (and somewhat easier for our concerns) problem, since basically by compactness, one just needs to approximate the function locally on a finite number of intervals. A 1986 reference survey discussing both approximation over the real line and over compacts is @cite_9 . Recently, over compact domains, the existence of universal ordinary differential equation @math of order @math has been established in @cite_13 : it is shown that for any @math , there exists a third order @math differential equation @math whose solutions are dense in @math . Notice that this is not obtained by explicitly stating such an order @math universal ordinary differential, and that this is a weaker notion of universality as solutions are only assumed to be arbitrary close over a compact domain and not all the real line. Order @math is argued to be a lower bound for Lipschitzian universal ODEs @cite_13 .
@cite_9 @cite_13
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In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to compute arbitrarily close piece-wise interval approximations. Even though this fragment is undecidable, we prove that there is a (possibly non-terminating) algorithm for checking satisfiability such that (1) whenever it terminates, it computes a correct answer, and (2) it always terminates when the input is robust. A formula is robust, if its satisfiability does not change under small perturbations. As a basic tool for our algorithm we use the notion of degree from the field of (differential) topology. Abstract We revisit the efficient approximation of functions by sums of exponentials or Gaussians in Beylkin and Monzon (2005) [16] to discuss several new results and applications of these approximations. By using the Poisson summation to discretize integral representations of e.g., power functions r − β , β > 0 , we obtain approximations with uniform relative error on the whole real line. Our approach is applicable to a class of functions and, in particular, yields a separated representation for the function e − x y . As a result, we obtain sharper error estimates and a simpler method to derive trapezoidal-type quadratures valid on finite intervals. We also introduce a new reduction algorithm for the case where our representation has an excessive number of terms with small exponents. As an application of these new estimates, we simplify and improve previous results on separated representations of operators with radial kernels. For any finite but arbitrary accuracy, we obtain new separated representations of solutions of Laplace's equation satisfying boundary conditions on the half-space or the sphere. These representations inherit a multiresolution structure from the Gaussian approximation leading to fast algorithms for the evaluation of the solutions. In the case of the sphere, our approach provides a foundation for a new multiresolution approach to evaluating and estimating models of gravitational potentials used for satellite orbit computations. A simple way to solve a partial differential equation in a non-rectangular domain @W is to embed the domain in a rectangle B and solve the problem (more easily) in the rectangle. To apply a Fourier spectral method on the rectangle, it is necessary to extend the inhomogeneous term in the PDE, f(x,y), to B in such a way that the extended function g(x,y) is periodic and infinitely differentiable, and yet is equal to f(x,y) everywhere in the irregular domain @W. If the boundary of @W, @[email protected], is defined as the zero isoline of a function @F(x,y), then a suitable extension is g(x,y)=f(x,y)H([email protected](x,y) constant) where H is a smoothed approximation to the step function which is defined in the main text. Abstract A universal ordinary differential equation C ∞ of order 3 is constructed here. The equation is universal in the sense that any continuous function on a real segment can be approximated by a solution of this equation with an arbitrary accuracy in uniform norm.