An extremal problem for even positive definite entire functions of exponential type

We consider an extremal problem for even positive definite entire functions of exponential type with zero mean with power weight on the semiaxis. This problem is related to the multidimensional Jackson-Stechkin theorem in the space L ₂(?ⁿ).     

Adaptive algorithm in irregular sampling of band-limited functions

This article obtains an adaptive algorithm in irregular sampling of band-limited functions, that is to say, important information on how the sampling density affects the performance of the algorithm is discovered. Moreover, the rate of convergence of the algorithm of the discrete model is improved.     

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

     

多元函数的最佳逼近递减的逼近阶

关于找一个必要与充分条件使得ωｉ１…ｋｉｍ（ｆ,１σｉ１,…,１σｉｍ）～Ｙσｉ１…σｉｍ（ｆ）,（σｉｊ→∞）成立的Ｔｉｍａｎ问题被解决。这条件是ωｋ１…ｋｉｍ（ｆ,δｉ１…δｉｍ）～ωｋｉ１＋１,…,ｋｉｍ＋１（ｆ,δｉ１…δｉｍ）,（δｉｊ→０）。     

The Levin-Golovin theorem for sobolev spaces

We obtain sufficient conditions for the basis property of the family of exponentials

On a theorem of Marcinkiewicz and Zygmund

Let {ϕn(x)} be an orthonormal system on the closed interval [0,1], and let ∥ϕ _n ∥_∞ ≤M _n . In 1937 Marcinkiewicz and Zygmund obtained an estimate of the norm inL _q [0,1] of the sum of the series ∑ _n=1 ^∞ c _n ϕ _n (x) under the condition that {M _n } is monotone increasing. In this paper it is shown that this condition cannot be discarded. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 386–390, March, 1998. The author wishes to thank V. I. Kolyada for setting the problem and for his permanent attention to this work.     

非有限带函数由Hermite型多元样本定理的重构

In this paper, we prove that under some restricted conditions, the non-bandiimited functions can be reconstructed by the multidimensional sampling theorem of Hermite type in the space of Lp（R^n）, 1 〈 p 〈 ∞.     

A sampling theorem with error estimation for S-transform

S-transform (ST) is an extension of wavelet transform (WT) and short-time Fourier transform (STFT). The ST is depended on a scalable and moving Gaussian window. It overcomes the low-resolution factor of STFT and tackles with a lack of phase in WT. The ST is a useful tool for signal processing and analysis. In this paper, the multiresolution analysis (MRA) related with ST is introduced and after that a sampling theorem associated with ST is proposed which is based on multiresolution subspace. Additionally, truncation and aliasing error generated due to the sampling process, are also derived. The theoretical determinations are exhibited and validated using simulation results.     

A sampling theorem related to the Wilson transform

In this paper, we establish a sampling theorem related to the nonterminating version of the Wilson polynomials and we give a generalization of the de Branges–Wilson integral.     

Carleson and sampling measures on Bernstein spaces on Siegel CR manifolds

In this paper, we introduce and study Carleson and sampling measures on Bernstein spaces on a class of quadratic CR manifold called Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to the given Siegel CR manifold are $L^p$-integrable with respect to a natural measure. For these spaces, we prove necessary and sufficient conditions for a Radon measure to be a Carleson or a sampling measure. We also provide sufficient conditions for sampling sequences.     

多元样本定理及混淆误差的估计

本文证明了多元指数型整函数的一个Ｍａｒｃｉｎｋｉｅｗｉｃａ型不等式，并由此证得了多元Ｗｈｉｔｔａｋｅｒ－Ｋｏｔｅｌｎｉｋｏｖ－Ｓｈａｎｎｏｎ型的样本定理，从而得到了多元Ｓｏｂｏｌｅｖ类上的混淆误差界的阶的精确估计。     

Hermite型多元样本定理及Sobolev类上混淆误差的估计

本文证明了Hermite型多元样本定理,并由此确定了Sobolev类上混淆误差阶的精确估计．     

多元凸函数的性质及其Hermite-Hadamard不等式

在多元凸函数的定义基础上,论述了多元凸函数的几种判定方法和一些性质,并证明了特殊区域上的Hermite-Hadamard不等式.     

Eigenvalues of periodic Sturm-Liouville problems by the Shannon-Whittaker sampling theorem

We are concerned with the computation of eigenvalues of a periodic Sturm-Liouville problem using interpolation techniques in Paley-Wiener spaces. We shall approximate the Hill discriminant by sampling a few of its values and then find its zeroes which are the square roots of the eigenvalues. Computable error estimates are provided together with eigenvalue enclosures.

     

Sampling theorem of Hermite type and aliasing error on the Sobolev class of functions

Denote by B _2σ,p (1 < p < ∞) the bandlimited class p-integrable functions whose Fourier transform is supported in the interval [−σ, σ]. It is shown that a function in B _2σ,p can be reconstructed in L _p(ℝ) by its sampling sequences {f (κπ / σ)} _κ∈ℤ and {f’ (κπ / σ)} _κ∈ℤ using the Hermite cardinal interpolation. Moreover, it will be shown that if f belongs to L _p ^r (ℝ), 1 < p < ∞, then the exact order of its aliasing error can be determined. Project supported by the Scientific Research Common Program of Beijing Municipal Commission of Education under grant number KM 200410009010 and by the Natural Science Foundation of China under grant number 10071006     

Multivariate Lukacs theorem

According to the celebrated Lukacs theorem, independence of quotient and sum of two independent positive random variables characterizes the gamma distribution. Rather unexpectedly, it appears that in the multivariate setting, the analogous independence condition does not characterize the multivariate gamma distribution in general, but is far more restrictive: it implies that the respective random vectors have independent or linearly dependent components. Our basic tool is a solution of a related functional equation of a quite general nature. As a side effect the form of the multivariate distribution with univariate Pareto conditionals is derived.     

Frames and sampling theorem

The sampling problem on a closed subspace V_o of L²(ℝ) is studied. For the regular sampling, a necessary and sufficient condition on the sampling theorem is presented and the sampling problem proposed by Walter is completely solved. Moreover, a representation formula for the derived function is given. The irregular sampling is studied. The results improve the ones of Walter and Liu’s. Project supported by the National Natural Science Foundation of China (Grant No. 16971047).     

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in . This orthonormal system is a generalization of the classical exponential system defining Fourier series.