scholarly journals Zeros of the Zak Transform of averaged totally positive functions

2017 ◽  
Vol 222 ◽  
pp. 55-63
Author(s):  
O.L. Vinogradov ◽  
A.Yu. Ulitskaya
Keyword(s):  
Zak Transform ◽  
Totally Positive ◽  
Positive Functions

scholarly journals Totally positive functions through nonstationary subdivision schemes

2007 ◽  
Vol 200 (1) ◽  
pp. 255-265 ◽  
Author(s):  
Costanza Conti ◽  
Laura Gori ◽  
Francesca Pitolli
Keyword(s):  
Subdivision Schemes ◽  
Totally Positive ◽  
Positive Functions

scholarly journals Zeros of the Wigner distribution and the short-time Fourier transform

2019 ◽  
Vol 33 (3) ◽  
pp. 723-744 ◽  
Author(s):  
Karlheinz Gröchenig ◽  
Philippe Jaming ◽  
Eugenia Malinnikova
Keyword(s):  
Fourier Transform ◽  
Bessel Functions ◽  
Wigner Distribution ◽  
Short Time Fourier Transform ◽  
Step Functions ◽  
Wigner Distributions ◽  
Hurwitz Polynomials ◽  
Totally Positive ◽  
Short Time ◽  
Positive Functions

AbstractWe study the question under which conditions the zero set of a (cross-) Wigner distribution W(f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson’s theorem for the positivity of the Wigner distribution and to Hardy’s uncertainty principle. We then construct a class of step functions S so that the Wigner distribution $$W(f,\mathbf {1}_{(0,1)})$$ W ( f , 1 ( 0 , 1 ) ) always possesses a zero $$f\in S \cap L^p$$ f ∈ S ∩ L p when $$p<\infty $$ p < ∞ , but may be zero-free for $$f\in S \cap L^\infty $$ f ∈ S ∩ L ∞ . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

scholarly journals Sign changes in linear combinations of derivatives and convolutions of Polya frequency functions

1979 ◽  
Vol 2 (1) ◽  
pp. 91-101
Author(s):  
Steven Nahmias ◽  
Frank Proschan
Keyword(s):  
Upper Bounds ◽  
Linear Combinations ◽  
Sign Changes ◽  
Variation Diminishing ◽  
Totally Positive ◽  
Positive Functions ◽  
Frequency Functions

We obtain upper bounds on the number of sign changes of linear combinations of derivatives and convolutions of Polya frequency functions using the variation diminishing properties of totally positive functions. These constitute extensions of earlier results of Karlin and Proschan.

scholarly journals Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

2017 ◽  
Vol 211 (3) ◽  
pp. 1119-1148 ◽  
Author(s):  
Karlheinz Gröchenig ◽  
José Luis Romero ◽  
Joachim Stöckler
Keyword(s):  
Gabor Frames ◽  
Sampling Theorems ◽  
Totally Positive ◽  
Shift Invariant Spaces ◽  
Positive Functions ◽  
Invariant Spaces

Crossing Properties of Mixture Distributions

1989 ◽  
Vol 3 (3) ◽  
pp. 355-366 ◽  
Author(s):  
Philip J. Boland ◽  
Frank Proschan ◽  
Y. L. Tong
Keyword(s):  
Mixture Distributions ◽  
Variation Diminishing ◽  
Gamma Distributions ◽  
Totally Positive ◽  
Variation Diminishing Property ◽  
Positive Functions ◽  
Family Of Distributions

Mixture distributions are a frequently used tool in modelling random phenomena. We consider mixtures of densities from a one-parameter exponenvial family of distributions. Using the tools of totally positive functions and the variation-diminishing property of such, we study the effect of sign-crossing properties of two mixing densities μ1 and μ2 on the resulting mixture distributions f1 and f2. The results enable us to make stochastic and variability cornparisons for binomial-beta, mixed Weibull, and mixed gamma distributions.

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