Exponential Box Splines Research Articles

The Inversion of Multiscale Convolution Approximation and Average of Distributions

We develop two kinds of inversion formulas of the multiscale convolution approximation which is defined by a convolution kernel φ. The inversion formulas are constructed by a convolution kernel ψ which is defined in terms of φ and has a vanishing moment of order one. A large class of generalized moving average approximations with B-splines, Box-splines and exponential Box-splines (EB-splines) as convolution kernels φ is included in the theory formulated in this paper. The average of distributions is considered, and correspondingly, the formulas related to the EB-splines are obtained from the δ-average.

Exponential Box-Like Splines on Nonuniform Grids

We generalize the exponential box spline by allowing it to have arbitrarily spaced knots in any of its directions and derive the corresponding recurrence and differentiation rules. The corresponding spline space is spanned by the shifts of finitely many such splines and contains the usual family of exponential polynomials. The (local) linear independence of the spanning set is equivalent to a geometric condition closely related to unimodularity.

On the Approximation Order of Principal Shift-Invariant Subspaces ofLp([formula omitted]d)

For 1⩽p⩽∞, sufficient conditions on the generators {φh}h>0are given which ensure that theh-dilates of the shift-invariant space generated byφhprovideLp-approximation of orderk>0. Examples whereφhis an exponential box spline or certain dilates of the Gaussiane−| · |2are considered; it is shown that our sufficient condition then provides an optimal lower bound on their approximation order.

Remarks on the linear independence of integer translates of exponential box splines

Following N. Sivakumar ( J. Approx. Theory 64 (1991) , 95–118), we study in this note the problem of the linear independence of the integer translates of an exponential box spline associated with a rational direction set.

Linear Independence of the Translates of an Exponential Box Spline

Linear Independence of the Translates of an Exponential Box Spline

Approximation by Piecewise Exponentials

A function is called an exponential if it is a linear combination of products of polynomials with pure exponentials. In this paper lower and upper bounds for families of spaces of piecewise exponentials are established. In particular, the exact $L_p $-approximation order $(1 \leq p \leq \infty )$ is found for a family $\{ {S_h } \}_{h > 0} $ of function spaces when each $S_h$ is generated by an exponential box spline and its multi-integer translates.

On The Convolution of a Box Spline With a Compactly Supported Distribution: Linear Independence for the Integer Translates

AbstractThe problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

Concerning the linear dependence of integer translates of exponential box splines

Let B ξ, λ be the exponential box spline associated with λ ϵ C n , and an s × n rational matrix with rank s and non-zero columns. Sufficient conditions are provided for the kernel space K(B Ξλ)≔ a:Z s→C: ∑ j∈Z s a(j)B Ξλ(·− j)=0 to be (i) trivial and (ii) finite dimensional. While these results extend the corresponding theorems known for integer matrices, the methods of proof are discernibly different.

Homogeneous polynomial splines

SynopsisWe construct functions which are piecewise homogeneous polynomials in the positive octant in three dimensions. These give a rich and elegant theory which combines properties of polynomial box splines see [6] and the references therein) with the explicit representation of simple exponential box splines [11], while enjoying complete symmetry in the three variables. By a linear transformation followed by a projection on suitable planes, one obtains piecewise polynomial functions of two variables on a mesh formed by three pencils of lines. The vertices of these pencils may be finite or one or two may be infinite, i.e. the corresponding pencils may comprise parallel lines. As a limiting case, all three vertices become infinite and one recovers polynomial box splines on a three-direction mesh.

On the Integer Translates of a Compactly Supported Function: Dual Bases and Linear Projectors

Given a multivariate compactly supported function $\phi $, linear projectors to the space $S(\phi )$ spanned by its integer translates are discussed here. These projectors are constructed with the aid of a dual basis for the integer translates of $\phi $, hence under the assumption that these translates are linearly independent. The main result shows that the linear functionals of the dual basis are local, hence making it possible to contruct local linear projectors onto $S(\phi )$. A scheme for the construction of such local projectors is then discussed for a general compactly supported function. In the second part of the paper these observations are applied to piecewise-polynomials and piecewise-exponentials to obtain a necessary and sufficient condition for a quasi interpolant to be a projector. The results of that part extend and refine recent constructions of dual bases and linear projectors for polynomial and exponential box splines.

Cardinal interpolation by symmetric exponential box splines on a three-direction mesh

We consider certain exponential box splines E on a three-direction mesh whose exponents satisfy a symmetry condition. It is shown, in particular, that given bounded data on the integer lattice in R2, there is a unique bounded combination of integer translates of E that interpolates the data. When all exponents are zero, this reduces to a result of de Boor, Höllig and Riemenschneider in [2]. Unlike the proof in [2] we use only elementary analysis and do not employ any computer calculations.

Local Approximation by Certain Spaces of Exponential Polynomials, Approximation Order of Exponential Box Splines, and Related Interpolation Problems

Local approximation order to smooth complex valued functions by a finite dimensional space $\mathcal {H}$, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space $\mathcal {H}$ is based here on the identification of its dual with a certain space $\mathcal {P}$ of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from $\mathcal {P}$, with interpolation data characterized by the structure of $\mathcal {H}$, and to construct bases of $\mathcal {P}$ corresponding to the interpolation problem.

Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems

Local approximation order to smooth complex valued functions by a finite dimensional space H \mathcal {H} , spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space H \mathcal {H} is based here on the identification of its dual with a certain space P \mathcal {P} of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from P \mathcal {P} , with interpolation data characterized by the structure of H \mathcal {H} , and to construct bases of P \mathcal {P} corresponding to the interpolation problem.

A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution

Given a multivariate compactly supported distributionϕ, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of\(\hat \varphi\)=the Fourier-Laplace transform ofϕ. The utility of the condition is demonstrated by several examples and applications, showing, in particular, that previous results on box splines and exponential box splines can be derived from this condition by a simple combinatorial argument.

Periodic exponential box splines on a three direction mesh

Multivariate “truncated Tchebycheff” functions which generalize the univariate Green's function on the one hand and the multivariate “truncated powers” on the other hand are constructed and analysed for the three direction mesh. For weight functions which are products of an exponential and a periodic function, the application of suitable divided differences to the corresponding truncated Tchebycheff functions yields a new type of box spline termed “periodic exponential.” This class of box splines contains the exponential (hence polynomial) box splines of the three direction mesh as special cases. It also extends the notion of the univariate periodic exponential Tchebycheffian B-splines.

Exponential box splines

In a recent paper by Nira Dyn and the author, univariate cardinal exponential B-splines are shown to have a representation similar to the wellknown box spline representation of the univariate cardinal polynomialB-splines. Motivated by this, we construct, for a set ofn directions inZs and a vector of constants λ Rn, an “exponential box spline” which has the same smoothness and support as the polynomial box spline, and is a positive piecewise exponential in its support. We derive recurrence relations for the exponential box splines which are simpler than those for the polynomial case. A relatively simple structure of the space spanned by the translates of an exponential box spline is obtained for λ in a certain open dense set ofRn—the “simple” λ. In this case, the characterization of the local independence of the translates and related topics, as well as the proofs involved, are quite simple when compared with the polynomial case (corresponding toλ = 0).

Translates of Exponential Box Splines and their Related Spaces

Translates of Exponential Box Splines and their Related Spaces

Translates of exponential box splines and their related spaces

Exponential box splines ( E B EB -splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space H \mathcal {H} spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space H {\mathbf {H}} of all entire functions spanned by the integer translates of an E B EB -spline. This investigation relies on a detailed description of H \mathcal {H} and its discrete analog S \mathcal {S} . The approach taken here is based on the observation that the structure of H \mathcal {H} is relatively simple when H \mathcal {H} is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex E B EB -splines. The first is when the Fourier transform of the E B EB -spline vanishes at some critical points, the second is when H \mathcal {H} cannot be embedded in S \mathcal {S} and the third is when H {\mathbf {H}} is a proper subspace of H \mathcal {H} . We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.