Limit theorems in spline approximation

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to Σ∆ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is hr−1. This rate is optimal in the sense that a function can be...

Best Quadratic Spline Approximation

We present a method for hierarchical data approximation using quadratic simplicial elements for domain decomposition and field approximation. Higher-order simplicial elements can approximate data better than linear elements. Thus, fewer quadratic elements are required to achieve similar approximation quality. We use quadratic basis functions and compute best quadratic simplicial spline approxim...

Parallel Spline Approximation

Three-monotone spline approximation

For r ≥ 3, n ∈ N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differences [x0, x1, x2, x3] f are nonnegative for all choices of distinct points x0, . . . , x3 in [a, b]), we construct a spline s of degree r and of minimal defect (i.e., s ∈ Cr−1[a, b]) with n −1 equidistant knots in (a, b), which is also 3-monotone and satisfies ‖ f − s‖L∞[a,b] ≤ cω...

Nonconventional Limit Theorems in Averaging

We consider ”nonconventional” averaging setup in the form dX(t) dt = ǫB ` X(t), ξ(q1(t)), ξ(q2(t)), ..., ξ(ql(t)) ́ where ξ(t), t ≥ 0 is either a stochastic process or a dynamical system (i.e. then ξ(t) = F x) with sufficiently fast mixing while qj(t) = αjt, α1 < α2 < ... < αk and qj , j = k+1, ..., l grow faster than linearly. We show that the properly normalized error term in the ”nonconventio...