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Approximation Theory and Applied Harmonic Analysis / Théorie de l'approximation et analyse harmonique
(Org: Rong-Qing Jia and/et Bin Han)

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DAVID AMUNDSEN, Carleton University, Ottawa, Ontario
Highly stable explicit evolution schemes based on the Padé Approximant

The numerical solution of time-dependent ordinary and partial differential equations presents a number of well known difficulties-including, possibly, severe restrictions on time-step sizes for stability in explicit schemes versus the need for solution of challenging, generally nonlinear systems of equations in implicit schemes. In joint work with Oscar Bruno (Caltech), we introduce a new class of explicit schemes based on the use of one-dimensional Padé Approximants. This approach, which is as simple and inexpensive per time-step as other explicit schemes, possesses properties of stability similar to those offered by implicit methods. We illustrate these properties and the significant gains in efficiency which may be achieved through application to notoriously stiff systems of ODEs and PDEs.

LEN BOS, University of Calgary
Heisenberg frames

We discuss the construction of frames in Cⁿ consisting of n² unit vectors with the property that their mutual inner products have constant modulus. This is joint, on-going work with S. Waldron of the University of Auckland.

MARTIN BROOKS, National Research Council
Simplification of topological piecewise monotone functions

The concept of piecewise monotonicity is extended to arbitrary continuous functions between topological spaces. Simplification is a special form of approximation in which selected portions of a function's piecewise monotone structure is either destroyed or preserved. Successive simplification provides a multiscale representation of the original function.

The generalization of piecewise monotonicity is based on the monotone-light factorization of analytic topology, which provides a "middle space" between a function's domain and range, reflecting the function's monotone structure. Two functions are "have the same shape" when there exists a certain homeorphism between their middle spaces. Every function's middle space is covered by maximal monotone connected sets; a function is piecewise monotone when the closures of the components of these sets' interiors are each monotone and are finite in number. A simplification is a certain kind of map between two function's middle spaces; simplification either preserves or anihilates monotone pieces. For example, a real-valued piecewise monotone function's middle space is a topological directed acyclic graph; simplification reduces the complexity of these graphs.

ALEX BRUDNYI, Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta  T2N 1N4
Remez-type inequalities for analytic functions

In the talk I will describe some multi-dimensional distributional inequalities for analytic functions. I will apply them to a version of Hilbert's 16th Problem on the number of limit cycles for a planar polynomial vector fields.

WEN CHEN, University of Alberta, Edmonton, Alberta
Accuracy analysis for the first order sigma-delta system

Based on experiments and numerical simulations, one believes that the time average squre error in the first order sigma-delta modulator behaves like O(l^-3) with respect to the sampling ratio l. This conjecture stands as an open problem for a long time. Combining tools from number theory, harmonic analysis, real analysis and complex analysis, this paper shows the conjecture in some reasonable sense.

SERGE DUBUC, Université de Montréal, Montréal, Québec
Scalar subdivision schemes and Hermite subdivision schemes

We extend the main results about convergence of uniform scalar subdivision schemes to more general subdivision schemes. In particular we provide two criteria of convergence which are available even for non-interpolating schemes. These criteria can be used for Hermite subdivision schemes since any Hermite subdivision scheme can be reduced to a scalar subdivision scheme. Even if the Hermite subdivision scheme is uniform, the scalar subdivision scheme is not.

JEAN PIERRE GABARDO, McMaster University, Hamilton, Ontario
Wavelet sets and self-similarity

A wavelet set is a subset K of Rⁿ with the property that the inverse Fourier transform of the characteristic function of K is a wavelet associated with an expansive dilation matrix A. Dai, Larson and Speegle (1997) have proved the existence of wavelet sets associated with any dilation matrix and many authors have,since then, provided explicit constructions of wavelet sets for various dilation matrices. In this talk, we will focus on integral dilation matrices and discuss how, in certain cases, one can construct wavelet sets built using self-similar sets associated with the matrix A^t and an associated complete set of digits. This construction provide interesting and ßimple" examples of wavelet sets in many situations. We will also point how the solution of our problem is related to number theoretical questions involving radix expansion using the given matrix A^t and the given set of digits. This talk is based on joint work with Xiaojiang Yu (McMaster University).

KIRILL KOPOTUN, University of Manitoba
Coconvex polynomial approximation

Suppose that a function f changes its convexity finitely many times on a finite interval. We are interested in properties of approximation of this functions by polynomials p_n which are "coconvex" with it (for example, if f is smooth, then this is equivalent to saying that p_n"(x) f"(x) ³ 0 for all x). Some recent developments in this area will be discussed.

DAVID J. LEEMING, University of Victoria
Real roots of the Bernoulli Polynomials-the ambiguous case

The Bernoulli polynomials can be defined by the generating function

text et-1 = ¥ å n=0  Bn ( x )  tn n! ,    |t| < 2p

A number of authors (e.g. Delange, Dilcher, Inkeri and Leeming) have studied the number and position of the real roots of B_n ( x). It is well known that B_2n+1 ( 0 ) = B_2n+1 ( 1) = 0 and B_2n (r_n ) = B_2n ( s_n)=0, n > 0, where 1/6 < r_n < 1/4 and s_n=1-r_n. Also, the real roots of B_n ( x) which are greater than 1 occur in pairs in the intervals m < x < m+1, m=1,2,¼,M, with the only exceptions being n=4k+1 with only one root in 1 < x < 2 and n=4k which has one or three roots when m=M.

We consider the four cases n=j mod 4, j=0,1,2,3 separately. For example, when n=0 mod 4, we approximate a normalized B_n( x) by a cosine function. Using asymptotically precise bounds we can determine the location of the largest real root and thus also the exact number of real roots. This yields a more precise method than that of Delange for determining the cases where there is a pair of roots in the interval [ M+1/2,M+1] . For 1 £ n £ 500, there are 17 such cases. These cases provide the only (unlikely) possibility for a double root of B_n(x).

Similar asymptotically precise bounds can be obtained in the other three cases using a sine or cosine function to approximate a normalized B_n(x). (This is joint work with Roderick Edwards.)

DANIEL LEMIRE, IIT-National Research Council of Canada, Fredericton New Brunswick  E3B 9W4
Quadratic and cubic 2-step subdivision schemes

Subdivision schemes provide local and smooth interpolation and lead to compactly supported wavelets. Multi-step subdivision schemes are a generalization of subdivision schemes where coarse scale guesses can be recorded and then corrected at later stages. They are more local than subdivision schemes for a given degree of polynomial reproduction. As an example, we show that 2-point and 3-point 2-step subdivision schemes can be respectively quadratic and cubic as well as continuous and differentiable.

QUN MO, University of Alberta, Edmonton, Alberta
Symmetric wavelet frames with high vanishing moments

In wavelet analysis, symmetry and high vanishing moments are two highly desirable properties of a wavelet system. Recently, Daubechies, Han, Ron and Shen and independently, Chui, He and Stöckler discovered a very interesting method to construct wavelet frames that are derived from any given pair of refinable functions. Using their method, one can construct pairs of compactly supported symmetric wavelet frames with high vanishing moments. Inspired by their method, Han and myself did some further study on the following topics: construction of pairs of dual multiwavelet frames, a family of symmetric tight wavelet frames derived from B-splines, a characterization of a tight wavelet frame with two compactly supported symmetric generators. I shall report all the results we have obtained. Some examples are provided.

FARAMARZ SAMAVATI, University of Calgary, Calgary, Alberta
Diagrammatic tools for generating biorthogonal multiresolutions

In a previous work we introduced a construction designed to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work introduces diagrams and diagram interactions to replace matrices and matrix multiplication. The diagrams we use are patterns of value-labeled nodes, one type of diagram corresponding to each row or column of one of the matrices of a biorthogonal system. All types of diagrams used in the construction are defined on a common mesh of the multiresolution.

IVAN SELESNICK, Polytechnic University, Brooklyn, New York, USA
Motion-based 3-D wavelet frames for video processing

The denoising of video data should take into account both temporal and spatial dimensions, however, separable 3-D transforms have artifacts that degrade their performance in applications. We describe the design and implementation of the non-separable 3-D dual-tree complex wavelet frame for video processing. We show that this expansive transform gives a motion-based multi-scale decomposition for video-each subband corresponds to motion in a specific direction and at a specific scale. Although the transform is non-separable, it is based on a mostly separable implementation (an efficient implementation). The development of this transform depends on the design of pairs of wavelet bases where the wavelet associated with the second basis is the Hilbert transform of the wavelet associated with the first basis,

yg(t) = H {yh(t)}.

REMI VAILLANCOURT, Départément de mathématiques, Université d'Ottawa, Ottawa Ontario  K1N 6N5
Smooth tight frame wavelets and image microanalysis in the Fourier domain

General results on microlocal analysis and tight frames in R² are summarized. To perform microlocal analysis of tempered distributions, orthogonal multiwavelets, whose Fourier transforms consist of characteristic functions of squares or sectors of annuli, are constructed in the Fourier domain and are shown to satisfy a multiresolution analysis with several choices of scaling functions. To have good localization in both the x and Fourier domains, redundant smooth tight wavelet frames, with frame bounds equal to one, called Parseval wavelet frames, are obtained in the Fourier domain by properly tapering the above characteristic functions. These nonorthogonal frame wavelets can be generated by two-scale equations from a multiresolution analysis. A natural formulation of the problem is by means of pseudodifferential operators. Singularities, which are added to smooth images, can be localized in position and direction by means of the frame coefficients of the filtered images computed in the Fourier domain. Using Plancherel's theorem, the frame expansion of the filtered images is obtained in the x domain. Subtracting this expansion from the scarred images restores the original images.

YANG WANG, Georgia Institute of Technology
On existence of Gabor basis

This talk concerns the following problem: Let L be a full rank lattice in R^2d with density D(L)=1, is it always possible to find function f(x) such that the Gabor system {e^2ppxf(x-q): (p,q) Î L} is an orthonormal basis for L²(R^d). It is well-known that for d=1 this problem has an affirmative answer. But it remianed unsolved in higher dimensions. I'll present some results, especially its connection to tiling.

TONY WARE, University of Calgary, Calgary, Alberta  T2N 1N4
A semi-Lagrangian wavelet method for convection-dominated PDEs

We describe a semi-Lagrangian method for dealing with convection-dominated partial differential equations, combined with a wavelet-Galerkin discretisation for the spatial variables. The inner-products for the Galerkin approach are computed with a special numerical quadrature that is exact for (products of) functions that are in an appropriate multiresolution subspace. We investigate the stability of the method, and describe an adaptive implementation of the approach.

JIE XIAO, Memorial University of Newfoundland, St. John's, Newfoundland  A1C 5S7
Capacitary strongtype inequalities for fractional order variations

This talk will show that there exists a capacitary strongtype inequality for the fractional order variation. Moreover, the existence can be effectively used in the study of domination principle (asymptotic formula for fractional order co-area), Sobolev type embedding, a priori estimation for certain Laplace equation, and their geometric inequalities.

SHUZHANG XU, Aristech at 628 Cedar Hill Drive, Allentown, Pennsylvania  18109, USA
Some simple analysis in ML, MAP and turbo decoding

We present some very simple analysis in the study of the most commonly used ML, MAP and turbo decoding in wireless communications. The purpose of these decoding schemes is to improve the reception quality of the transmitted digital information with extra processing. Our analysis gives some interpretations and guidelines to the design and understanding of practical decoding schemes. We will cover the following topics:

(1)  SNR dependency of ML and MAP decoder windowing techniques

(2)  Path metric and performance bound improvement due to extrinsic information

(3)  Simple quality indexes and virtual SNR calculation

(4)  ARQ schemes with Yamamoto-Itoh type indexes

(5)  Adaptive iterative decoding schemes and high speed decoding schemes

Some numerical results will be presented to support our analysis. Our straightforward analysis shows also how practical communication problems can be analyzed with very simple mathematics. In short, this note is a brief summary of [5]. This is a joint work with Wayne Stark.

References

[1]

C. Berrou et al, Near Shannon limit error-correcting coding and decoding: Turbo codes. IEEE Inter. Conf. On Comm., May, 1993, 1064-1070.

[2]

L. Bahl et al, Optimal decoding of linear codes for minimizing symbol error rate. IEEE Trans. Inform. Theory 20(1974), 284-287.

[3]

J. Hagenauer et al, Iterative decoding of binary block and convolutional codes. IEEE Trans. Inform. Theory 42(1996), 429-445.

[4]

A. Viterbi and J. Omura, Principles of digital communication and coding. McGraw-Hill, 1979

[5]

S. Xu and W. Stark, Extrinsic information impact on ML and MAP decoding of convolutional codes. submitted to IEEE Trans Comm.

YUESHENG XU, West Virginia University
Greedy algorithms and tree wavelet approximation

We develop a constructive greedy scheme (CGS) that realizes tree wavelet approximations. We define a function class F_p^a which contains the functions whose global error generated by CGS have a prescribed decreasing rate. Embedding properties related to these function classes are investigated. We also show that the tree wavelet approximation based on CGS will also have a decreasing error rate O(N^-a) for functions in some Besov spaces B^a_qq.

PING ZHOU, St. Francis Xavier University, Antigonish, Nova Scotia  B2G 2W5
Multivariate Padé approximants to some special functions

Padé approximants for univariate functions and their applications are well-known. Multivariate Padé approximants are relatively new and we have very few explicit constructions. In this talk, I will introduce the definition of multivariate Padé approximants and discuss some explicit constructions of multivariate Padé approximants to some special functions such as the Appell functions.