On the Best Approximation by Ridge Functions in the Uniform Norm

We consider the best approximation of some function classes by the manifold M _n consisting of sums of n arbitrary ridge functions. It is proved that the deviation of the Sobolev class W _p ^r,d from the manifold M _n in the space L _q for any 2≤ q≤ p≤∈fty behaves asymptotically as n ^-r/(d-1) . In particular, we obtain this asymptotic estimate for the uniform norm p=q=∈fty . January 10, 2000. Date revised: March 1, 2001. Date accepted: March 12, 2001.     

Variational Principles and Sobolev-Type Estimates for Generalized Interpolation on a Riemannian Manifold

The purpose of this paper is to study certain variational principles and Sobolev-type estimates for the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation on a closed (i.e., no boundary), compact, connected, orientable, m -dimensional C ^∞ Riemannian manifold , with C ^∞ metric g _ij . The rate of approximation can be more fully analyzed with rates of approximation given in terms of Sobolev norms. Estimates on the rate of convergence for generalized Hermite and other distributional interpolants can be obtained in certain circumstances and, finally, the constants appearing in the approximation order inequalities are explicit. Our focus in this paper will be on approximation rates in the cases of the circle, other tori, and the 2 -sphere. April 10, 1996. Dates revised: March 26, 1997; August 26, 1997. Date accepted: September 12, 1997. Communicated by Ronald A. DeVore.     

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.     

On Multivariate Adaptive Approximation

Recently, A. Cohen, R. A. DeVore, P. Petrushev, and H. Xu investigated nonlinear approximation in the space BV (R ² ). They modified the classical adaptive algorithm to solve related extremal problems. In this paper, we further study the modified adaptive approximation and obtain results on some extremal problems related to the spaces V _σ,p ^r (R ^d ) of functions of ``Bounded Variation" and Besov spaces B ^α (R ^d ). November 23, 1998. Date revised: June 25, 1999. Date accepted: September 13, 1999.     

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φ_v) _v=0 ^/r−1 of vector refinement equations. The space spanned by the translates of φ_v can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸_v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.     

Factoring wavelet transforms into lifting steps

This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z⁻¹])=E(n;R[z, z⁻¹])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. Research Tutorial Acknowledgements and Notes. Page 264.     

On Existence and Weak Stability of Matrix Refinable Functions

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation where P is an r×r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997.     

Approximation by p -Faber polynomials in the weighted Smirnov class Ep( G,ω ) and the Bieberbach polynomials

Let G\subset C be a finite domain with a regular Jordan boundary L . In this work, the approximation properties of a p -Faber polynomial series of functions in the weighted Smirnov class E ^p (G,ω) are studied and the rate of polynomial approximation, for f∈ E ^p ( G,ω) by the weighted integral modulus of continuity, is estimated. Some application of this result to the uniform convergence of the Bieberbach polynomials π _n in a closed domain \overline G with a smooth boundary L is given. February 25, 1999. Date revised: October 20, 1999. Date accepted: May 26, 2000.     

Sparse Components of Images and Optimal Atomic Decompositions

Recently, Field, Lewicki, Olshausen, and Sejnowski have reported efforts to identify the ``Sparse Components' of image data. Their empirical findings indicate that such components have elongated shapes and assume a wide range of positions, orientations, and scales. To date, sparse components analysis (SCA) has only been conducted on databases of small (e.g., 16 by 16) image patches and there seems limited prospect of dramatically increased resolving power. In this paper, we apply mathematical analysis to a specific formalization of SCA using synthetic image models, hoping to gain insight into what might emerge from a higher-resolution SCA based on n by n image patches for large n but a constant field of view. In our formalization, we study a class of objects \cal F in a functional space; they are to be represented by linear combinations of atoms from an overcomplete dictionary, and sparsity is measured by the ℓ <sup>p</sup> -norm of the coefficients in the linear combination. We focus on the class \cal F = \sc Star <sup>α</sup> of black and white images with the black region consisting of a star-shaped set with an α -smooth boundary. We aim to find an optimal dictionary, one achieving the optimal sparsity in an atomic decomposition uniformly over members of the class \sc Star <sup>α</sup> . We show that there is a well-defined optimal sparsity of representation of members of \sc Star <sup>α</sup> ; there are decompositions with finite ℓ <sup>p</sup> -norm for p > 2/(α+1) but not for p < 2/(α+1) . We show that the optimal degree of sparsity is nearly attained using atomic decompositions based on the wedgelet dictionary. Wedgelets provide a system of representation by elements in a dyadically organized collection, at all scales, locations, orientations, and positions. The atoms of our atomic decomposition contain both coarse-scale dyadic ``blobs,' which are simply wedgelets from our dictionary, and fine-scale ``needles,' which are differences of pairs of wedgelets. The fine-scale atoms used in the adaptive atomic decomposition are highly anisotropic and occupy a range of positions, scales, and locations. This agrees qualitatively with the visual appearance of empirically determined sparse components of natural images. The set has certain definite scaling properties; for example, the number of atoms of length l scales as 1/l , and, when the object has α -smooth boundaries, the number of atoms with anisotropy \approx A scales as \approx A ^α-1 . August 16, 1999. Date revised: April 24, 2000. Date accepted: April 4, 2000.     

An Adaptive Compression Algorithm in Besov Spaces

Given a function f on [0,1] and a wavelet-type expansion of f , we introduce a new algorithm providing an approximation $\tilde f of f with a prescribed number D of nonzero coefficients in its expansion. This algorithm depends only on the number of coefficients to be kept and not on any smoothness assumption on f . Nevertheless it provides the optimal rate D ^-α of approximation with respect to the L _q -norm when f belongs to some Besov space B ^α _p,∈fty whenever α>(1/p-1/q) ⁺ . These results extend to more general expansions including splines and piecewise polynomials and to multivariate functions. Moreover, this construction allows us to compute easily the metric entropy of Besov balls. June 21, 1996. Dates revised: April 9, 1998; October 14, 1998. Date accepted: October 20, 1998.     

Weighted Maximum Over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants

Let a≥ 0 , ɛ >0 . We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set Here P is an arbitrary polynomial of degree ≤ n . We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal \left{ [n/n]\right} _n=1 ^∞ sequence provides good approximation on almost one-eighth of the circles centre 0 , and the \left{ [2n/n]\right} _n=1 ^∞ sequence on almost one-quarter of such circles. July 18, 2000. Date revised: . Date accepted: April 19, 2001.     

Cesaro Summability with Respect to Two-Parameter Walsh Systems

 The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H ^p to L ^p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T _α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T _α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L ^p spaces to L ^p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way. (Received 20 April 2000; in revised form 25 September 2000)     

Bernstein—Szegő's Theorem for Sobolev Orthogonal Polynomials

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product where ρ ₀ and ρ ₁ are weights which satisfy Szegő's condition, supported on a smooth Jordan closed curve or arc. December 14, 1997. Date revised: September 21, 1998. Date accepted: November 16, 1998.     

Nonlinear Methods of Approximation

Abstract. Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classification, and so on. The standard problem in this regard is the problem of m -term approximation where one fixes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in some given norm (or metric). We are also interested in stable algorithms for finding good or near best approximations using m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms), and adaptive basis selection. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. With this motivation, our recent work and the current activity focuses on nonlinear approximation both in the classical form of m -term approximation (where several important problems remain unsolved) and in the form of highly nonlinear approximation where a theory is only now emerging.     

On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms ||⋅|| _Hs([0,1]) for s∈ (-0.824926,2.5) . In particular, the multiwavelets form Riesz bases for L ₂ ([0,1]) . February 2, 1998. Date revised: February 19, 1999. Date accepted: March 5, 1999.     

Approximation Orders of FSI Spaces in L 2 (R d )

A second look at the authors' [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant subspace S(Φ) of L ₂ (R ^d ) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators of the subspace. Further, when the generators satisfy a certain technical condition, then, under the mild assumption that the set of 1-periodizations of the generators is linearly independent, such a space is shown to provide approximation order k if and only if contains a ψ (necessarily unique) satisfying . The technical condition is satisfied, e.g., when the generators are at infinity for some ρ>k+d . In the case of compactly supported generators, this recovers an earlier result of Jia [J1], [J2]. March 19. 1996. Date revised: September 6, 1996.     

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.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

   Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).