A New Class of Curves of Rational B-Spline Type

A new class of rational parametrization has been developed and it was used to generate a new family of rational k functions B-splines which depends on an index α ∈ ]−∞ , 0[ ∪ ]1 , +∞[. This family of functions verifies, among other things, the properties of positivity, of partition of the unit and, for a given degree k, constitutes a true basis approximation of continuous functions. We loose, however, the regularity classical optimal linked to the multiplicity of nodes, which we recover in the asymptotic case, when α → ∞. The associated B-splines curves verify the traditional properties particularly that of a convex hull and we see a certain “conjugated symmetry” related to α. The case of open knot vectors without an inner node leads to a new family of rational Bezier curves that will be separately, object of in-depth analysis.

On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series

Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function f by general means τλn;A(f) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function H(t) ≥ 0 and via entries of a matrix whose indices form a strictly increasing sequence of positive integers λ := {λ(n)}∞n=1.

A note on approximation of continuous functions on normed spaces

Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions, which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from a complex separable normed space, admitting a separating $*$-polynomial, can be uniformly approximated by $*$-analytic functions.

On the nonsymmetric approximation of continuous functions in the integral metric

In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes $H^\omega[a,b]$ is proved. Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P. Korneichuk, and the exact estimate for the best one-sided approximation obtained by V.G. Doronin and A.A. Ligun.