10 research outputs found
Cardinal interpolation with polysplines on annuli
AbstractCardinal polysplines of order p on annuli are functions in C2p-2Rn⧹0 which are piecewise polyharmonic of order p such that Δp-1S may have discontinuities on spheres in Rn, centered at the origin and having radii of the form ej, j∈Z. The main result is an interpolation theorem for cardinal polysplines where the data are given by sufficiently smooth functions on the spheres of radius ej and center 0 obeying a certain growth condition in j. This result can be considered as an analogue of the famous interpolation theorem of Schoenberg for cardinal splines
Thin plate splines for transfinite interpolation at concentric circles
We propose a new method for constructing a polyspline on annuli, i.e. a C 2 surface on ℝ2 \ {0}, which is piecewise biharmonic on annuli centered at 0 and interpolates smooth data at all interface circles. A unique surface is obtained by imposing Beppo Levi conditions on the innermost and outermost annuli, and one additional restriction at 0: either prescribing an extra data value, or asking that the surface is non-singular. We show that the resulting Beppo Levi polysplines on annuli are in fact thin plate splines, i.e. they minimize Duchon's bending energy
Radially symmetric thin plate splines interpolating a circular contour map
Profiles of radially symmetric thin plate spline surfaces minimizing the
Beppo Levi energy over a compact annulus have been
studied by Rabut via reproducing kernel methods. Motivated by our recent
construction of Beppo Levi polyspline surfaces, we focus here on minimizing the
radial energy over the full semi-axis . Using a -spline
approach, we find two types of minimizing profiles: one is the limit of Rabut's
solution as and (identified as a
`non-singular' -spline), the other has a second-derivative singularity and
matches an extra data value at . For both profiles and , we establish the -approximation order in
the radial energy space. We also include numerical examples and obtain a novel
representation of the minimizers in terms of dilates of a basis function.Comment: new figures and sub-sections; new Proposition 1 replacing old
Corollary 1; shorter proof of Theorem 4; one new referenc
Polyharmonic functions of infinite order on annular regions
Polyharmonic functions f of infinite order and type {\tau} on annular regions
are systematically studied. The first main result states that the
Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of
infinite order and type 0 can be extended to analytic functions on the complex
plane cut along the negative semiaxis. The second main result gives a
constructive procedure via Fourier-Laplace series for the analytic extension of
a polyharmonic function on annular region A(r_{0},r_{1}) of infinite order and
type less than 1/2r_{1} to the kernel of the harmonicity hull of the annular
region. The methods of proof depend on an extensive investigation of Taylor
series with respect to linear differential operators with constant
coefficients.Comment: 32 page
Bernstein operators for exponential polynomials
Let be a linear differential operator with constant coefficients of order
and complex eigenvalues . Assume that the set
of all solutions of the equation is closed under complex
conjugation. If the length of the interval is smaller than , where M_{n}:=\max \left\{| \text{Im}% \lambda_{j}| :j=0,...,n\right\}
, then there exists a basis %, , of the space with
the property that each has a zero of order at and a zero of
order at and each is positive on the open interval
Under the additional assumption that and
are real and distinct, our first main result states that there exist points and positive numbers %,
such that the operator \begin{equation*}
B_{n}f:=\sum_{k=0}^{n}\alpha_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies
, for The second main result
gives a sufficient condition guaranteeing the uniform convergence of
to for each .Comment: A very similar version is to appear in Constructive Approximatio
Cardinal interpolation with polysplines on annuli
Cardinal polysplines of order p on annuli are functions in C 2p−2 (R n \{0}) which are piecewise polyharmonic of order p such that ∆ p−1 S may have discontinuities on spheres in R n, centered at the origin and having radii of the form e j,j ∈ Z. The main result is an interpolation theorem for cardinal polysplines where the data are given by sufficiently smooth functions on the spheres of radius e j and center 0 obeying a certain growth condition in |j|. This result can be considered as an analogue of the famous interpolation theorem of Schoenberg for cardinal splines. Key words: Cardinal splines, Schoenberg interpolation theorems, L−splines, cardinal spline interpolation, spherical harmonics, polyharmonic functions i