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LISSA: Padua Points and Lissajous Sampling
The Padua Points are the
first
known example of optimal points for
total degree polynomial interpolation
in two variables,
with a Lebesgue constant increasing like log square of the degree.
They have been discovered and studied by our group during some collaboration
periods
at the University
of Padua
with Len Bos (Calgary), Shayne Waldron (Auckland)
and Yuan Xu (Eugene).
Lagrange interpolation at the Padua points has been recently used
in several scientific and technological applications, for example in
Computational Chemistry (the
Fun2D subroutine of the
CP2K
simulation package for Molecular Dynamics, see
paper), in Image
Processing (algorithms for
image
retrieval
by colour indexing), in
Materials Science (Modelling of Composite Layered Materials,
see paper), in
Mathematical Statistics (Copula Density Estimation, see abstract by L.
Qu, p. 67), in Quantum Physics (Quantum State Tomography, see paper);
moreover,
it has been
added in the
Chebfun2 package.
One of the key features of the Padua Points is that they lie on
a particular Lissajous curve.
The more general topic of multivariate Polynomial Approximation on
Lissajous Curves
turned out to be of interest in the emerging field of
Magnetic Particle Imaging (MPI)
(see, e.g., some recent publications below and the activities of the
scientific network
MathMPI). Lissajous sampling seems to be relevant also in the field
of
Atomic Force Microscopy (AFM).
Papers
 Rhodonea curves as sampling
trajectories for spectral
interpolation on the unit disk
arXiv preprint 1812.00437  W. Erb
Constr. Approx. 53 (2021), 281318

Spectral filtering for the reduction of the Gibbs phenomenon for
polynomial approximation methods on Lissajous curves with applications
in MPI
preprint  S. De Marchi, W. Erb and F. Marchetti
Dolomites Res. Notes Approx. DRNA 10 (2017), 128137

Lissajous sampling and spectral filtering in
MPI applications: the reconstruction
algorithm for reducing the Gibbs phenomenon
preprint  S. De Marchi, W. Erb and F. Marchetti
2017 International Conference on Sampling Theory and Applications
(SampTA), IEEE Xplore Digital Library, DOI: 10.1109/SAMPTA.2017.8024375

Polynomial approximation on Lissajous curves in the dcube
preprint  L. Bos, S. De Marchi and M. Vianello
Appl. Numer. Math. 116 (2017), 4756

A Simple Recipe for Modelling a dcube by Lissajous
curves
L. Bos
Dolomites Res. Notes Approx. DRNA 10 (2017), 14
 Trivariate
polynomial approximation on Lissajous curves
preprint  L. Bos, S. De Marchi and M. Vianello
IMA J. Numer. Anal. 37 (2017), 519541
 On certain
multivariate Vandermonde determinants whose variables separate
preprint  S. De Marchi and K. Usevich
Linear Algebra Appl. 449 (2014), 1727

Padua2DM: fast interpolation and cubature at the Padua points in
Matlab/Octave
preprint  M. Caliari, S. De Marchi, A. Sommariva and M. Vianello
Numer. Algorithms 56 (2011), 4560

On the Vandermonde Determinant of Padualike Points
L. Bos, S. De Marchi and S. Waldron
Dolomites Research Notes on Approximation 2 (2009), 115

A numerical code for fast interpolation and cubature at
the Padua points
preprint  M. Caliari, S. De Marchi, A. Sommariva and M. Vianello
Proceedings of the 9th CMMSE (2009), Vol. I,
218228

Algorithm 886:
Padua2D: Lagrange Interpolation at
Padua
Points on Bivariate
Domains
preprint  M. Caliari, S. De Marchi and M. Vianello
ACM Trans. Math. Software 353 (2008)

Nontensorial
ClenshawCurtis cubature
preprint  A. Sommariva, M. Vianello and R. Zanovello
Numer. Algorithms 49 (2008), 409427

Bivariate Lagrange
interpolation at the Padua
points:
computational aspects
preprint  M. Caliari, S. De Marchi and M. Vianello
J. Comput. Appl. Math. 221 (2008), 284292

Bivariate Lagrange
interpolation at the
Padua points: the
ideal theory approach
preprint  L. Bos, S. De Marchi, M. Vianello and Y. Xu
Numer. Math. 108 (2007), 4357

Bivariate
Lagrange interpolation at the
Padua points: the
generating curve approach
preprint  L. Bos, M. Caliari, S. De Marchi, M. Vianello and Y. Xu
J. Approx. Theory 143 (2006), 1525

Bivariate polynomial
interpolation on the square at new nodal sets
preprint  M. Caliari, S. De Marchi and M. Vianello
Appl. Math. Comput. 165/2 (2005), 261274
Codes
 Padua2D
(Fortran 77 code for
interpolation
at
Padualike points
on rectangles, triangles and ellipses); the code is also in the Netlib
by M. Caliari, S. De Marchi and
M. Vianello (see paper) 
ACM Trans. Math.
Software 353 (2008)
note: a variant has been used in
Fun2D of the
CP2K
simulation package for molecular dynamics
by M. Guidon, J. Hutter and J. VandeVondele (see
paper)
 Padua2DM
(a Matlab/Octave code for interpolation and cubature at the Padua
points); the code is also in the Netlib
by M. Caliari, S. De Marchi, A. Sommariva and M.
Vianello (see paper) 
Numer. Algorithms 56
(2011)

hyperlissa
(Matlab code for hyperinterpolation on a Lissajous curve of the cube)
with a
demo
by S. De Marchi and M.
Vianello (see paper)

RDisk
(Matlab codes for spectral interpolation on Rhodonea nodes
of the unit disk)
by W. Erb (see paper)
Thirdparty codes