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One outcome is a simple proof that for $g_{m \alpha , n \beta}$ to span $L^2,$ the lattice $(m \alpha , n \beta )$ must have at least unit density.
      
Applications of generalized perron trees to maximal functions and density bases
      
Convergence and summability of Gabor expansions at the Nyquist density
      
It is well known that Gabor expansions generated by a lattice of Nyquist density are numerically unstable, in the sense that they do not constitute frame decompositions.
      
In 1994, Kotz, Ostrovskii and Hayfavi [10] carried out a detailed investigation of analytic and asymptotic properties of the density of the distribution for the symmetric case θ=0.
      
Under the appropriate definition of sampling density D?, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D?≥1.
      
On Generating Tight Gabor Frames at Critical Density
      
In this article we deal with the Arov-Grossman functional model to describe all the solutions of the Covariance Extension Problem for q-variate stationary stochastic processes and we find the density that maximizes the Burg Multivariate Entropy.
      
With this correspondence, the density that maximizes the Burg Multivariate Entropy corresponds to the function H\equiv0.
      
Density, Overcompleteness, and Localization of Frames.
      
Density, Overcompleteness, and Localization of Frames.
      
Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified.
      
New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame.
      
History and Evolution of the Density Theorem for Gabor Frames
      
The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis.
      
We consider tight Gabor frames (h,a=1,b=1) at critical density with h of the form Z-1(Zg/|Zg|).
      
VEGF, microalbuminuria, HbA1c, creatinine, triglycerides, total cholesterol, fasting serum glucose, high-density lipoprotein (HDL), and low-density lipoprotein (LDL) were measured.
      
In this paper, we construct the EB estimators of θ by using the kernel estimation of multivariate density function and its partial derivatives.
      
As applications, we discuss the local influence of small perturbations of normal-gamma prior density in linear model and investigate local prior influence from the predictive view.
      
Asymptotics of the residuals density estimation in nonparametric regression underm(n)-dependent sample
      
 

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