On Condition Numbers Associated with Radial-Function Interpolation

On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian

“…Let λ > 0 be fixed. Suppose that (x j : j ∈ Z) is a real sequence satisfying (8) and (9) for some positive numbers q and Q. Let A = A λ be the bi-infinite matrix whose entries are given by A( j, k) = e −λ(x j −x k ) 2 , j, k ∈ Z.…”

“…In particular we do not assume in Theorem 6.7(iii) that (x j : j ∈ Z) is a Riesz-basis sequence. However, it is not without interest to note that the main convergence theorems obtained in Section 4 do not hold for data sites which merely satisfy the quasi-uniformity conditions (8) and (9). For example, let X := Z \ {0}, and let f be the bandlimited function given by f (x) := sin(π x)/(π x), x ∈ R. As f (x j ) = 0 for every x j ∈ X , I λ ( f ) is identically zero, so it is manifest that I λ ( f ) does not converge to f .…”

Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms

“…To make the statement clearer, we give some more additional assumptions on the Mercer kernel K(x, y) dened by (2).…”

“…Let t ⊂ X be discrete nite sets satisfying Assumption 4, K t (x) be a Mercer kernel dened by (2) and satisfying Assumptions 12 and X K t (x) dμ(x) = 1. K t (φ, x) dened by (11) satises Assumption 3.…”

On the stability of meshless symmetric collocation for boundary value problems

“…To this end we use a simplified version of results for the classical interpolation problem (see for example [12,13,15,17]). We will show that there is a trade-off principle concerning convergence and stability, which is much worse than the one known from classical interpolation [17].…”