共查询到20条相似文献,搜索用时 15 毫秒
1.
Gaurav Thakur 《Journal of Functional Analysis》2011,260(8):2283-2299
We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties. 相似文献
2.
Approximation of signals from local averages 总被引:4,自引:0,他引:4
This work is concerned with approximation of a signal from local averages. It improves a result of Butzer and Lei [P.L. Butzer, J. Lei, Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal. 4 (2000) 245–255]. 相似文献
3.
We introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions.We use a generalization of a representation theorem due to Carathéodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. For a given accuracy ε, selecting an eigenvalue close to ε yields an approximate quadrature with that accuracy. To compute its weights and nodes, we present a new fast algorithm.These new quadratures can be used to approximate and integrate bandlimited functions, such as prolate spheroidal wave functions, and essentially bandlimited functions, such as Bessel functions. We also develop, for a given precision, an interpolating basis for bandlimited functions on an interval. 相似文献
4.
Yoel Shkolnisky Mark Tygert Vladimir Rokhlin 《Applied and Computational Harmonic Analysis》2006,21(3):413-420
Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae. 相似文献
5.
6.
C. Vinti 《Results in Mathematics》1998,34(1-2):32-55
Here we give a survey on recent results in approximation theory obtained by a group of mathematicians of the University of Perugia, inspired by the work of Professor Butzer. We collect some results concerning linear integral operators in various settings. 相似文献
7.
Isaac Z. Pesenson Meyer Z. Pesenson 《Journal of Mathematical Analysis and Applications》2010,366(1):137-152
We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory. 相似文献
8.
Karlheinz Gröchenig Gitta Kutyniok Kristian Seip 《Journal of Functional Analysis》2008,255(7):1831-1850
We derive necessary conditions for sampling and interpolation of bandlimited functions on a locally compact abelian group in line with the classical results of H. Landau for bandlimited functions on Rd. Our conditions are phrased as comparison principles involving a certain canonical lattice. 相似文献
9.
In this paper, we introduce a Durrmeyer‐type generalization of q‐Bleimann, Butzer, and Hahn operators based on q‐integers and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. We also compute rates of statistical convergence of these q‐type operators by means of the modulus of continuity and Lipschitz‐type maximal function, respectively. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
10.
We determine the best asymptotic constant in an estimation of the second central moment of the Bleimann, Butzer and Hahn operators L n yielding some new results concerning their rate of convergence. In particular, we find $$\lim_{n\in\mathbb{N}}\sup_{0 < x < \infty}\frac{n\,L_{n}((\cdot-x)^{2} ;x)}{x(1+x)^{2}}$$ . 相似文献
11.
Robert S. Strichartz 《Journal of Fourier Analysis and Applications》2016,22(5):1157-1173
On a bipartite graph G we consider the half sampling problem of uniquely recovering a function from its values on the even vertices, under the appropriate half bandlimited assumption with respect to a Laplacian on the graph. We discuss both finite and infinite graphs, give the appropriate definition of “half bandlimited” that involves splitting the mid frequency, and give an explicit solution to the problem. We discuss in detail the example of a regular tree. We also consider a related sampling problem on graphs that are generated by edge substitution. 相似文献
12.
Mehmet Ali Özarslan Hüseyin Aktuğlu 《Mathematical Methods in the Applied Sciences》2015,38(9):1937-1944
In this paper, we prove a certain Korovkin type approximation theorem by introducing new test functions. We introduce the non‐tensor Balázs type Bleimann, Butzer and Hahn operators and give the approximation property by using this new Korovkin theorem. Furthermore, we obtain the rate of convergence of these operators by means of modulus of continuity. Finally, we state the multivariate version of the abovementioned Korovkin type theorem. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
13.
Götz E. Pfander 《Journal of Fourier Analysis and Applications》2013,19(3):612-650
Sampling and reconstruction of functions is a fundamental tool in science. We develop an analogous sampling theory for operators whose Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for in this sense bandlimited operators and show that our results generalize both, the classical sampling theorem, and the fact that a time-invariant operator is fully determined by its impulse response. 相似文献
14.
Gabriella DiBlasio 《Semigroup Forum》1998,57(3):359-377
tA )t≥0 on X. We denote by XA(θ,p), for 0 < θ < 1, the interpolation spaces of X and D(A) introduced by Berens and Butzer in [1]. These spaces play an important
role in the approximation theory ( [2] ) as well as in the study of the abstract parabolic equation u′ (t) = Au(t) + f(t) ( [3] ). It has been proved in [6] that these spaces are meaningful (and still enjoy relevant properties) also in the case
θ = 0. In this paper we continue the study of [6] and prove new interesting properties of these spaces. 相似文献
15.
Mario DeFranco 《Journal of Difference Equations and Applications》2013,19(9):1101-1120
We present an analytic extension of the unsigned Stirling numbers of the first kind that is in a certain sense unique in its coincidence with the Stirling polynomials. We examine and compare our extension to previous extensions of (signed) Stirling numbers of the first kind given by Butzer et al. (2007, J. Difference Equ. Appl., 13) and of the unsigned numbers given by Adamchik (1997, J. Comput. Appl. Math., 79). We also see a connection to the Riemann zeta function. 相似文献
16.
We state a result about the local saturation of sequences of linear operators that preserve the sign of the k-th derivative of the functions. We apply it to the well known approximation operators of Bernstein, Szász–Mirakjan, Meyer–König and Zeller, and Bleimann, Butzer and Hahn. 相似文献
17.
Indranil SenGupta Bo Sun Weisheng Jiang Goong Chen Maria C. Mariani 《Journal of Fourier Analysis and Applications》2012,18(1):182-210
The concentration problem of maximizing signal strength of bandlimited and timelimited nature is important in communication
theory. In this paper we consider two types of concentration problems for the signals which are bandlimited in disjoint frequency-intervals,
which constitute a band-pass filter. For the first type the problem is to determine which members of L
2(−∞,∞) lose the smallest fraction of their energy when first timelimited and then bandlimited. For the second type the problem
is to determine which bandlimited signals lose the smallest fraction of their energy when restricted to a given time interval.
For both types of problems, basic theoretical properties and numerical algorithms for solution and convergence theorems are
given. Orthogonality properties of analytically extended eigenfunctions over L
2(−∞,∞) are also proved. Numerical computations are carried out which corroborate the theory. Relationship between eigenvalues
of these two types of problems is also established. Several properties of eigenvalues of both types of problems are proved. 相似文献
18.
Tianxiao HE 《数学研究及应用》2012,32(6):631-640
Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, k-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [16]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed,which extend the corresponding results about the Stirling numbers shown in [21] to the defined Stirling functions. 相似文献
19.
Gyorgy Gdt 《分析论及其应用》2003,19(1):55-75
The aim of this paper is to prove the following theorem concerning the term by term differentiation the-orem of Walsh-Kaczmarz series. Let (ck) be a decreasing real sequence withare integrable functions and f(x) is a. e. dyadic (or Butzer and Wagner) differentiate withThe function Kk means the kth Walsh-Kaczmarz function. 相似文献
20.
We investigate the ability of deep deep rectified linear unit (ReLU) networks to approximate multivariate functions. Specially, we establish the approximation error estimate on a class of bandlimited functions; in this case, ReLU networks can overcome the “curse of dimensionality.” 相似文献