Universal approximation by translates of fundamental solutions of elliptic equations
Vassili Nestoridis, Yiorgos‐Sokratis Smyrlis · Analysis · 2011
In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by universal series of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie on a prescribed surface outside of – Ω, known as the pseudo-boundary . The domains under consideration satisfy a rather mild boundary regularity requirement, namely, the segment condition. We study approximations with respect to the norms of the spaces C ℓ ( – Ω)and we establish the existence of universal series. Analogous results are obtainable with respect to the norms of Hölder spaces C ℓ , ν ( – Ω). The sequence a = { a n } n ∈ ℕ of coefficients of the universal series may be chosen in ∩ p > 1 l p (ℕ) but it can not be chosen in l 1 (ℕ).