A Distributional Approach to Asymptotics

On the support of tempered distributions

“…The spaces of test functions D, E, and S and the corresponding spaces of distributions are well-known [5]. We shall also need the distribution space K , the dual of K; the space K plays a fundamental role in the theory of summability of distributional evaluations [5].…”

“…We shall also need the distribution space K , the dual of K; the space K plays a fundamental role in the theory of summability of distributional evaluations [5].…”

“…We shall follow [3,5] for the notions related to Cesàro behavior of distributions. If f ∈ D (R n ) and α ∈ R is not a negative integer, we say that f is bounded by |x| α in the Cesàro sense for |x| large, and write…”

“…Cesàro summability. We can also consider Cesàro summability by spherical means [5,Section 6.8]. Summability by spherical means can actually be reduced to summability in one variable since using polar coordinates, x =rω, r ≥ 0, ω ∈ S, we obtain…”

Hierarchical quadrature for multidimensional singular integrals – Part II

“…If −1 < α < 0, the kernel function is improperly integrable. If α −1 the integrals are strongly singular and I has to be interpreted as a partie finie integral in the sense of Hadamard [4,6,7]. Our assumptions are explained for…”