In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrab...In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.展开更多
Signals are often of random character since they cannot bear any information if they are predictable for any time t, they are usually modelled as stationary random processes .On the other hand, because of the inertia ...Signals are often of random character since they cannot bear any information if they are predictable for any time t, they are usually modelled as stationary random processes .On the other hand, because of the inertia of the measurement apparatus, measured sampled values obtained in practice may not be the precise value of the signal X(t) at time tk (k∈Z), but only local averages of X(t) near tk. In this paper, it is presented that a wide (or weak ) sense stationary stochastic process can be approximated by generalized sampling series with local average samples.展开更多
Generalized sampling in a shift invariant subspace V of L2（R） is considered. A function f in V is processed with different filters Lm and then one tries to reconstruct f from the samples L^mf（j＇k）. We develop a t...Generalized sampling in a shift invariant subspace V of L2（R） is considered. A function f in V is processed with different filters Lm and then one tries to reconstruct f from the samples L^mf（j＇k）. We develop a theory of how to do this in the case when V possesses a shift invariant frame. Special attention is paid to the question： How to obtain dual frames with compact support？展开更多
In this paper an asymptotic formula of Voronovskaja type for a multivariate extension of the Kantorovich generalized sampling series is given. Moreover a quantitative version in terms of some moduli of smoothness is e...In this paper an asymptotic formula of Voronovskaja type for a multivariate extension of the Kantorovich generalized sampling series is given. Moreover a quantitative version in terms of some moduli of smoothness is established. Finally some particular examples of kernels are discussed, as the Bochner-Riesz kernel and the multivariate splines.展开更多
文摘In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.
基金National Natural Science Foundation of China (No60572113,No10501026) and Liuhui Center for Applied Mathematics
文摘Signals are often of random character since they cannot bear any information if they are predictable for any time t, they are usually modelled as stationary random processes .On the other hand, because of the inertia of the measurement apparatus, measured sampled values obtained in practice may not be the precise value of the signal X(t) at time tk (k∈Z), but only local averages of X(t) near tk. In this paper, it is presented that a wide (or weak ) sense stationary stochastic process can be approximated by generalized sampling series with local average samples.
文摘Generalized sampling in a shift invariant subspace V of L2（R） is considered. A function f in V is processed with different filters Lm and then one tries to reconstruct f from the samples L^mf（j＇k）. We develop a theory of how to do this in the case when V possesses a shift invariant frame. Special attention is paid to the question： How to obtain dual frames with compact support？
文摘In this paper an asymptotic formula of Voronovskaja type for a multivariate extension of the Kantorovich generalized sampling series is given. Moreover a quantitative version in terms of some moduli of smoothness is established. Finally some particular examples of kernels are discussed, as the Bochner-Riesz kernel and the multivariate splines.
