10 research outputs found

    Cardinal interpolation with polysplines on annuli

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    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

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    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

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    Profiles of radially symmetric thin plate spline surfaces minimizing the Beppo Levi energy over a compact annulus R1rR2R_{1}\leq r\leq R_{2} 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 0<r<0<r<\infty. Using a LL-spline approach, we find two types of minimizing profiles: one is the limit of Rabut's solution as R10R_{1}\rightarrow0 and R2R_{2}\rightarrow\infty (identified as a `non-singular' LL-spline), the other has a second-derivative singularity and matches an extra data value at 00. For both profiles and p[2,]p\in\left[ 2,\infty\right] , we establish the LpL^{p}-approximation order 3/2+1/p3/2+1/p 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

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    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

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    Let LL be a linear differential operator with constant coefficients of order nn and complex eigenvalues λ0,...,λn\lambda_{0},...,\lambda_{n}. Assume that the set UnU_{n} of all solutions of the equation Lf=0Lf=0 is closed under complex conjugation. If the length of the interval [a,b][ a,b] is smaller than π/Mn\pi /M_{n}, where M_{n}:=\max \left\{| \text{Im}% \lambda_{j}| :j=0,...,n\right\} , then there exists a basis pn,kp_{n,k}%, k=0,...nk=0,...n, of the space UnU_{n} with the property that each pn,kp_{n,k} has a zero of order kk at aa and a zero of order nkn-k at b,b, and each % p_{n,k} is positive on the open interval (a,b).(a,b) . Under the additional assumption that λ0\lambda_{0} and λ1\lambda_{1} are real and distinct, our first main result states that there exist points a=t0<t1<...<tn=b% a=t_{0}<t_{1}<...<t_{n}=b and positive numbers α0,..,αn\alpha_{0},..,\alpha_{n}%, 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 Bneλjx=eλjxB_{n}e^{\lambda_{j}x}=e^{\lambda_{j}x}, for j=0,1.j=0,1. The second main result gives a sufficient condition guaranteeing the uniform convergence of BnfB_{n}f to ff for each fC[a,b]f\in C[ a,b] .Comment: A very similar version is to appear in Constructive Approximatio

    Cardinal interpolation with polysplines on annuli

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    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
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