Quasi-interpolation with translates of a function having noncompact support

We establish a result related to a theorem of de Boor and Jia [1]. Their theorem, in turn, corrected and extended a result of Fix and Strang [5] concerning controlled approximation. In our result, the approximating functions are not required to have compact support, but satisfy instead conditions on their behavior at . Our theorem includes some recent results of Jackson [6] and is closely related to the work of Buhmann [2].Communicated by Carl de Boor     

On multivariate approximation by integer translates of a basis function

Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted” fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasi-interpolation is addressed first. The analysis makes an essential use of the structure of the generalized Fourier transform of the basis function. In contrast with spline theory, polynomial reproduction is not sufficient for the derivation of exact order of convergence by dilated quasi-interpolants. These convergence orders are established by a careful and quite involved examination of the decay rates of the basis function. Furthermore, it is shown that the same approximation orders are obtained with quasi-interpolants defined on a bounded domain. Supported in part by the United States under contract No. DAAL-87-K-0030, and by Carl de Boor’s Steenbock Professorship, University of Wisconsin-Madison.     

Fundamental sets of continuous functions on spheres

LetS ^m andS ^∞ denote the unit spheres inR m+1 ande ², respectively. We look for functionsf inC[?1, 1] such that the family of functionsx →f(<s,v>) is fundamental in the spaceC(S ^m). Herev runs overS ^m. There is a similar question forC(S ^∞), when this space is given the topology of uniform convergence on compact sets.     

The polynomials in the linear span of integer translates of a compactly supported function

Algebraic facts about the space of polynomials contained in the span of integer translates of a compactly supported function are derived and then used in a discussion of the various quasi-interpolants from that span.     

A scheme for interpolation by Hankel translates of a basis function

This work discusses interpolation of complex-valued functions defined on the positive real axis $I$ by certain special subspaces, in a variational setting that follows the approach of Light and Wayne [W. Light, H. Wayne, Spaces of distributions, interpolation by translates of a basis function and error estimates, Numer. Math. 81 (1999) 415–450]. The set of interpolation points will be a subset $\{a_1,\dots ,a_n\}$ of $I$ and the interpolants will take the form $u\left(x\right)=\sum _{i=1}^n{\alpha }_i\left({\tau }_{a_i}?\right)\left(x\right)+\sum _{j=0}^{m?1}{\beta }_j{p}_{\mu ,j}\left(x\right)(x\in I)\text{,}$ where $\mu \ge ?1/2,?$ is a complex function defined on $I$ (the so-called basis function), $p_{\mu ,j}\left(x\right)=x^{2j+\mu +1/2}(j\in {\mathbb{Z}}_{+},0\le j\le m?1)$ is a Müntz monomial, ${\tau }_z(z\in I)$ denotes the Hankel translation operator of order $\mu$, and ${\alpha }_i,{\beta }_j(i,j\in {\mathbb{Z}}_{+},1\le i\le n,0\le j\le m?1)$ are complex coefficients. An estimate for the pointwise error of these interpolants is given. Some numerical examples are included.     

The derivative of a continuous nonconstant function

Using elementary ideas and techniques, we prove (Theorem 2) that for a nonconstant differentiable function , a < b, the set S(f) = {x ε [a, b] : f'(x) ≠ 0} cannot be negligible. This result remains valid if f fails to be differentiable on a countable subset of [a, b].     

Analyticity on translates of a Jordan curve

Let be a domain in which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate vertically to get where is such that . We prove that if is a continuous function on such that for each , the function has a continuous extension to which is holomorphic on , then is holomorphic on .

     

A remark on translates of a box spline

We prove an express theorem on the stability of integer translates of a box spline and clear proposition 4. 10. 2 in [6] for box spline case. Also, we provide a way to construct a linear projection from L_p to the space of integer translates of a box spline. Consequently, the equivalence among the stability, global linear independence and local linear independence of translates of a box spile follows. Partially supported by ARO contract No. DAAL 4-87-0025     

The fundamentality of sets of ridge functions

Summary We investigate the fundamentality of the set of all continuous ridge functions in the spaceC( ⁿ ) as well as inC(X) for a general Banach space,X. Both positive and negative results are obtained. Necessary and sufficient conditions for the fundamentality are given for certain sets of ridge functions inC( ⁿ ).     

Spaces of distributions, interpolation by translates of a basis function and error estimates

Summary. Interpolation with translates of a basis function is a common process in approximation theory. The most elementary form of the interpolant consists of a linear combination of all translates by interpolation points of a single basis function. Frequently, low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it is often the solution of a variational problem. In this paper we concentrate on developing a wide variety of spaces for which a variational theory is available. For each of these spaces, we show that there is a natural choice of basis function. We also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates. Received December 10, 1996 / Revised version received August 29, 1997     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

The local Hölder function of a continuous function

This work focuses on the local Hölder exponent as a measure of the regularity of a function around a given point. We investigate in detail the structure and the main properties of the local Hölder function (i.e., the function that associates to each point its local Hölder exponent). We prove that it is possible to construct a continuous function with prescribed local and pointwise Hölder functions outside a set of Hausdorff dimension 0.     

Direct form seminorms arising in the theory of interpolation by translates of a basis function

In the error analysis of the process of interpolation by translates of a single basis function, certain spaces of functions arise naturally. These spaces are defined with respect to a seminorm which is given in terms of the Fourier transform of the function. We call this an indirect seminorm. In certain well‐understood cases, the seminorm can be rewritten trivially in terms of the function, rather than its Fourier transform. We call this a direct seminorm. The direct form allows better error estimates to be obtained. In this paper, we shown how to rewrite most of the commonly arising indirect form seminorms in direct form, and begin a little of the analysis required to obtain the improved error estimates. This revised version was published online in June 2006 with corrections to the Cover Date.     

Point transversals to translates of a trapezoid

Anm-transversal to a family of convex sets in the plane is anm-point set which intersects every members of the family. One of Grübaum’s conjectures says that a planar family of translates of a convex compact set has a 3-transversal provided that any two of its members intersect. Recently the conjecture has been proved affirmatively (see [4]). In the present paper we provide a different and straightforward proof for the conjecture for the family of translates of a closed trapezoid in the plane and give several concrete 3-transversals.     

Integration of a zonal function related to the local harmonic analysis on spheres

In studying local harmonic analysis on the sphere Sⁿ, R.S. Strichartz introduced certain zonal functions ϕ₂(d(x, y)) which satisfy the equation , where Δ_z is the Laplace operator and δ_−y the Dirac measure. The explicit expression of the constant a (λ) is given by R.S. Strichartz in the case that n is odd. Appyling the Apéry identity, we show in this paper that for n even, where w_n-1 is the surface area of S^n-1, . The author's research was supported by a grant from NSFC.