We prove the existence of series ∑anψn, whose coefficients (an) are in ∩p>1ℓp and whose terms (ψn) are translates by rational vectors in double-struckRd of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener’s algebra W(C0,ℓ1), Cb(Rd), C0(Rd), Lp(Rd), for every p∈[1,∞), and the space of measurable functions. Applying this theory to particular situations, we establish approximations by such series to solutions of the heat and Laplace equations as well as to probability density functions.

Abstract. We investigate the existence of solutions to systems of N differential equations representing connections between minima of potentials with several equal depths in R N . Using variational techniques and in particular a method introduced in [AF] we first prove such existence for N ≥ 2 and two minima. Dealing next with symmetric potentials corresponding to free bulk energies in crystals we establish existence for N ≥ 2 in various cases of more than two minima. Finally we obtain a sufficient condition establishing existence of connections to not necessarily symmetric potentials for arbitrary N and three minima.

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Link to this article: http://journals.cambridge.org/abstract_S0305004107000515 How to cite this article: VANGELIS STEFANOPOULOS (2008). Uniform approximation by universal series on arbitrary sets. AbstractBy considering a tree-like decomposition of an arbitrary set we prove the existence of an associated series with the property that its partial sums approximate uniformly all elements in a relevant space of bounded functions. In a topological setting we show that these sums are dense in the space of continuous functions, hence in turn any Borel measurable function is the almost everywhere limit of an appropriate sequence of partial sums of the same series. The coefficients of the series may be chosen in c 0 , or in a weighted p with 1 < p < ∞, but not in the corresponding weighted 1 .

In this paper an abstract condition is given yielding universal series defined by sequences a = {a j } ∞ j=1 in ∩p>1 p but not in 1 . We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann ζ-function, or a fundamental solution of a given elliptic operator in R ν with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in R ν simultaneously with respect to all σ-finite Borel measures in R ν . Stronger results are obtained by using universal Dirichlet series.