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Perfect splines and Hermite-Birkhoff interpolation
No full textAbstractFor a function ƒ0 in the Sobolev space W ∞[a,b], let F(ƒ0) be the set of all functions ƒ in W∞[a,b] satisfying the interpolation condition ƒ(j)(xi) = ƒ0(j)(xi) ∀ (i,j) with eij = 1, where a = x1 < x2 < … < xk = b and E = ∥ eij ∥ki = 1, m−1j = 0 is an incidence matrix. We investigate existence and extremal properties of perfect splines in F(ƒ0) under certain conditions on E
New Bounds on the Zeros of Spline Functions
No full textAbstractWe show that, subject to a certain condition, the number of zeros of a spline function is bounded by the number of strong sign changes in its sequence of B-spline coefficients. By writing a general spline function as a sum of functions which satisfy the given condition, we can deduce known bounds on zeros and sign changes and can show that the number of zeros of any spline function is bounded by the number of weak sign changes in its sequence of B-spline coefficients, where the zero count is stronger than that previously used
Characterising Pairs of Refinable Splines
No full textAbstractA complete characterisation is given, in terms of Fourier transforms, of pairs of refinable univariate spline functions, with knots at the integers, whose integer translates form a Riesz basis
Interpolatory Hermite Spline Wavelets
No full textAbstractWavelets are constructed comprising spline functions with multiple knots. These wavelets have certain derivatives vanishing at the integers, in an analogous manner to the B-splines of Schoenberg and Sharma related to cardinal Hermite interpolation
Two Ways to Construct a Smooth Piecewise Rational Curve
No full textAbstractTwo approaches to constructing piecewise rational curves are compared and contrasted. One involves projecting a piecewise polynomial curve whose polynomial segments are joined according to given totally positive connection matrices. The other, which is discussed in more detail, constructs the Bézier points of the rational segments by successively cutting corners of a given control polygon. The smoothness and total positivity properties of the resulting curves are also discussed
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