g=ptpfun(L,w) g=ptpfun(L,w,width)
| 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 |
| 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:
where \(m`=`numel(w)\)\(\geq 2\). 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.
K. Gröchenig and J. Stöckler. Gabor frames and totally positive functions. Duke Math. J., 162(6):1003--1031, 2013.
S. Bannert, K. Gröchenig, and J. Stöckler. 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. [ DOI | http ]
T. Kloos. Gabor frames total-positiver funktionen endlicher ordnung. Master's thesis, University of Dortmund, Dortmund, Germany, 2012.