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. January 10, 1996. Date revised: December 9, 1997. Date accepted: March 18, 1998.     

Averages Using Translation Induced by Laguerre and Jacobi Expansions

Approximation by averages of the generalized translation induced by Laguerre and Jacobi expansions will be shown to satisfy a strong converse inequality of type B with the appropriate K -functional. April 9, 1998. Date revised: February 22, 1999. Date accepted: March 5, 1999.     

Interpolation by Polynomials and Radial Basis Functions on Spheres

The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics. These interpolation processes take place in ``spherical caps' whose size is controlled by the fill distance, and the important aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest. March 27, 1997. Dates revised: March 19, 1998; August 5, 1999. Date accepted: December 15, 1999.     

Simple Cubature Formulas with High Polynomial Exactness

We study cubature formulas for d -dimensional integrals with arbitrary weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree, the number of knots depends on the dimension in an order-optimal way. The cubature formulas are universal: the order of convergence is almost optimal for two different scales of function spaces. The construction is simple: a small number of arithmetical operations is sufficient to compute the knots and the weights of the formulas. August 25, 1997. Date revised: December 3, 1998. Date accepted: March 3, 1999.     

Regularity of Irregular Subdivision

We study the smoothness of the limit function for one-dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural generalization of the four-point scheme introduced by Dubuc and Dyn, Levin, and Gregory, we show that, under some geometric restrictions, the limit function is always C ¹ ; under slightly stronger restrictions we show that the limit function is almost C ² , the same regularity as in the regularly spaced case. May 27, 1997. Date revised: March 10, 1998. Date accepted: March 28, 1998.     

On the Norm of the Fourier—Gegenbauer Projection in Weighted L p Spaces

We extend the results of Pollard [4] and give asymptotic estimates for the norm of the Fourier—Gegenbauer projection operator in the appropriate weighted L _p space. In particular, we settle the question of whether the projection is bounded for p=(2λ+1)/λ and p=(2λ+1)/(λ+1) , where λ is the index for the family of Gegenbauer polynomials under consideration. March 19, 1997. Date revised: June 3, 1998. Date accepted: August 1, 1998.     

Dependence Relations Among the Shifts of a Multivariate Refinable Distribution

Refinable functions are an intrinsic part of subdivision schemes and wavelet constructions. The relevant properties of such functions must usually be determined from their refinement masks. In this paper, we provide a characterization of linear independence for the shifts of a multivariate refinable vector of distributions in terms of its (finitely supported) refinement mask. March 14, 1998. Dates revised: February 3, 1999 and August 6, 1999. Date accepted: November 16, 1999.     

Greedy Algorithms with Regard to Multivariate Systems with Special Structure

The question of finding an optimal dictionary for nonlinear m -term approximation is studied in this paper. We consider this problem in the periodic multivariate (d variables) case for classes of functions with mixed smoothness. We prove that the well-known dictionary U ^d which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthonormal dictionaries. Next, it is established that for these classes near-best m -term approximation, with regard to U ^d , can be achieved by simple greedy-type (thresholding-type) algorithms. The univariate dictionary U is used to construct a dictionary which is optimal among dictionaries with the tensor product structure. June 22, 1998. Date revised: March 26, 1999. Date accepted: March 22, 1999.     

Biorthogonal Wavelet Expansions

This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular, we address the close connection of this issue with stationary subdivision schemes. Date received: May 20, 1995. Date revised: March 2, 1996.     

The Whittaker Function M κ ,μ as a Function of κ

The dependence of the Whittaker function M _{κ, μ} (z) on the parameter κ is considered. A convergent expansion in ascending powers and an asymptotic expansion in descending powers of κ are discussed. Some properties of the coefficients of the convergent expansion are shown. February 14, 1997. Date revised: March 5, 1998. Date accepted: March 19, 1998.     

On the Degree of Approximation by Manifolds of Finite Pseudo-Dimension

The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes. A class of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation width = which measures the worst-case approximation error over all functions by the best manifold of pseudo-dimension n . In this paper we obtain tight upper and lower bounds on ρ _n (W ^r,d _p , L _q ) , both being a constant factor of n ^-r/d , for a Sobolev class W ^r,d _p , . As this is also the estimate of the classical Alexandrov nonlinear n -width, our result proves that approximation of W ^r,d _p by the family of manifolds of pseudo-dimension n is as powerful as approximation by the family of all nonlinear manifolds with continuous selection operators. March 12, 1997. Dates revised: August 26, 1997, October 24, 1997, March 16, 1998, June 15, 1998. Date accepted: June 25, 1998.     

Multifractal Formalism for Self-Similar Functions Under the Action of Nonlinear Dynamical Systems

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{?}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis. October 1, 1996. Date revised: May 13, 1997. Date re-revised: January 10, 1998. Date accepted: February 27, 1998.     

Extreme Bloch Functions and Summation Methods for Bloch Functions

In this paper we construct Bloch functions F for which the set {z | e sup _{|ζ| < 1} |F'(ζ)| ( 1 - |ζ| ² ) = |F'(z)| ( 1 - |z| ² )} is an analytic Jordan curve tangential to the unit disk in some points. It is proved that, using such functions, we can derive analogs to the Taylor expansion for Bloch functions in cases where the Taylor expansion does not converge. October 15, 1997. Date revised: March 12, 1998. Date accepted: June 18, 1998.     

Note on the Approximation of Powers of the Distance in Two-Dimensional Domains

Although Newman's trick has been mainly applied to the approximation of univariate functions, it is also appropriate for the approximation of multivariate functions that are encountered in connection with Green's functions for elliptic differential equations. The asymptotics of the real-valued function on a ball in 2-space coincides with that for an approximation problem in the complex plane. The note contains an open problem. May 17, 1999. Date revised: October 20, 1999. Date accepted: March 17, 2000.     

A Strong Converse Inequality for Gamma-Type Operators

We give a strong converse inequality of type A in the usual sup-norm for a noncentered gamma operator L _t ^* , providing at the same time upper and lower constants. This operator, which does not preserve smooth functions, is connected with real Laplace transforms and Poisson mixtures. We use a probabilistic approach based on the representation of L _t ^* in terms of gamma processes. October 15, 1997. Date revised: September 14, 1998. Date accepted: October 7, 1998.     

Weighted Polynomial Inequalities with Doubling and A ∞ Weights

We consider weighted inequalities such as Bernstein, Nikolskii, Remez, etc., inequalities under minimal assumptions on the weights. It turns out that in most cases this mimimal assumption is the doubling condition. Sometimes, however, as for the Remez and Nikolskii inequalities, one needs the slightly stronger A _∈ fty condition. We shall consider both the trigonometric and the algebraic cases. August 20, 1997. Date revised: April 19, 1998. Date accepted: May 26, 1998.     

The Finite Element Method on the Sierpinski Gasket

For certain classes of fractal differential equations on the Sierpinski gasket, built using the Kigami Laplacian, we describe how to approximate solutions using the finite element method based on piecewise harmonic or piecewise biharmonic splines. We give theoretical error estimates, and compare these with experimental data obtained using a computer implementation of the method (available at the web site http://mathlab.cit.cornell.edu/\sim gibbons). We also explain some interesting structure concerning the spectrum of the Laplacian that became apparent from the experimental data. March 29, 2000. Date revised: March 6, 2001. Date accepted: March 21, 2001.     

More on Comonotone Polynomial Approximation

The main achievement of this paper is that we show, what was to us, a surprising conclusion, namely, twice continuously differentiable functions in (0,1) (with some regular behavior at the endpoints) which change monotonicity at least once in the interval, are approximable better by comonotone polynomials, than are such functions that are merely monotone. We obtain Jackson-type estimates for the comonotone polynomial approximation of such functions that are impossible to achieve for monotone approximation. July 7, 1998. Date revised: May 5, 1999. Date accepted: July 23, 1999.     

Restricted Nonlinear Approximation

We introduce a new form of nonlinear approximation called restricted approximation . It is a generalization of n -term wavelet approximation in which a weight function is used to control the terms in the wavelet expansion of the approximant. This form of approximation occurs in statistical estimation and in the characterization of interpolation spaces for certain pairs of L _p and Besov spaces. We characterize, both in terms of their wavelet coefficients and also in terms of their smoothness, the functions which are approximated with a specified rate by restricted approximation. We also show the relation of this form of approximation with certain types of thresholding of wavelet coefficients. March 31, 1998. Date accepted: January 28, 1999.     

Convergence Acceleration of Taylor Sections by Convolution

Taylor sections S _n (f) of an entire function f often provide easy computable polynomial approximants of f . However, while the rate of convergence of (S _n (f)) _n is nearly optimal on circles around the origin, this is no longer true for other plane sets as, for example, real compact intervals. The aim of this paper is to construct for certain families of (entire) functions sequences of polynomial approximants which are computable with essentially the same effort as Taylor sections and which have a better rate of convergence on some parts of the plane. The resulting method may be applied, for example, to (modified) Bessel functions, to confluent hypergeometric functions, or to parabolic cylinder functions. October 2, 1997. Date revised: March 12, 1998. Date accepted: April 28, 1998.