


Approximation Theory and Applied Harmonic Analysis / Théorie
de l'approximation et analyse harmonique (Org: RongQing Jia and/et Bin Han)
 DAVID AMUNDSEN, Carleton University, Ottawa, Ontario
Highly stable explicit evolution schemes based on
the Padé Approximant

The numerical solution of timedependent ordinary and partial
differential equations presents a number of well known
difficultiesincluding, possibly, severe restrictions on timestep
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
onedimensional Padé Approximants. This approach, which is as simple
and inexpensive per timestep 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 C^{n} consisting of n^{2} unit
vectors with the property that their mutual inner products have
constant modulus. This is joint, ongoing 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
monotonelight 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 realvalued 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
Remeztype inequalities for analytic functions

In the talk I will describe some multidimensional 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 sigmadelta system

Based on experiments and numerical simulations, one believes that the
time average squre error in the first order sigmadelta 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
noninterpolating 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 selfsimilarity

A wavelet set is a subset K of R^{n} 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
selfsimilar 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 Polynomialsthe ambiguous case

The Bernoulli polynomials can be defined by the generating function

te^{xt}
e^{t}1

= 
¥ å
n=0

B_{n} ( x ) 
t^{n}
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}=1r_{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, IITNational Research Council of Canada,
Fredericton New Brunswick E3B 9W4
Quadratic and cubic 2step subdivision schemes

Subdivision schemes provide local and smooth interpolation and lead to
compactly supported wavelets. Multistep 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 2point and 3point 2step 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 Bsplines, 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 tensorproduct surfaces. For mesh surfaces of nontensor
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
valuelabeled 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
Motionbased 3D wavelet frames for video processing

The denoising of video data should take into account both temporal and
spatial dimensions, however, separable 3D transforms have artifacts
that degrade their performance in applications. We describe the design
and implementation of the nonseparable 3D dualtree complex wavelet
frame for video processing. We show that this expansive transform
gives a motionbased multiscale decomposition for videoeach subband
corresponds to motion in a specific direction and at a specific scale.
Although the transform is nonseparable, 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,
 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
R^{2} 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 twoscale 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^{2ppx}f(xq): (p,q) Î L} is an orthonormal basis for
L^{2}(R^{d}). It is wellknown 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 semiLagrangian wavelet method for convectiondominated
PDEs

We describe a semiLagrangian method for dealing with
convectiondominated partial differential equations, combined with a
waveletGalerkin discretisation for the spatial variables. The
innerproducts 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 coarea), 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 YamamotoItoh 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 errorcorrecting coding
and decoding: Turbo codes. IEEE Inter. Conf. On Comm., May, 1993, 10641070.
 [2]
 L. Bahl et al, Optimal decoding of linear codes for
minimizing symbol error rate. IEEE Trans. Inform. Theory 20(1974),
284287.
 [3]
 J. Hagenauer et al, Iterative decoding of binary block and
convolutional codes. IEEE Trans. Inform. Theory 42(1996), 429445.
 [4]
 A. Viterbi and J. Omura, Principles of digital communication
and coding. McGrawHill, 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
wellknown. 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.

