PTPFUN Sampled, periodized totally positive function of finite type
Usage: g=ptpfun(L,w)
g=ptpfun(L,w,width)
Input parameters:
L : Window length.
w : Vector of reciprocals w_j=1/delta_j in Fourier representation of g*
width: Integer stretching factor for the essential support of g
Output parameters:
g : The periodized totally positive function.
PTPFUN(L,w) computes samples of a periodized totally positive
function of finite type >=2 with weights w for a system of length L.
The Fourier representation of the continuous TP function is given as:
m
ghat(f) = prod (1+2pijf/w(i))^(-1),
i=1
where m=numel(w)>= 2. The samples are obtained by discretizing
the Zak transform of the function.
w controls the function decay in the time domain. More specifically
the function decays as exp(max(w)x) for x->infty and exp(min(w)x)
for x->-infty assuming w contains both positive and negative
numbers.
PTPFUN(L,w,width) additionally stretches the function by a factor of
width.
References:
K. Groechenig and J. Stoeckler. Gabor frames and totally positive
functions. Duke Math. J., 162(6):1003--1031, 2013.
S. Bannert, K. Groechenig, and J. Stoeckler. Discretized Gabor frames of
totally positive functions. Information Theory, IEEE Transactions on,
60(1):159--169, 2014.
T. Kloos and J. Stockler. Full length article: Zak transforms and gabor
frames of totally positive functions and exponential b-splines. J.
Approx. Theory, 184:209--237, Aug. 2014. [1]http ]
T. Kloos. Gabor frames total-positiver funktionen endlicher ordnung.
Master's thesis, University of Dortmund, Dortmund, Germany, 2012.
References
1. http://dx.doi.org/10.1016/j.jat.2014.05.010
Url: http://ltfat.github.io/doc/fourier/ptpfun.html
See also: dgt, ptpfundual, gabdualnorm, normalize.
Package: ltfat