Oblique Projections and Sampling Problems

In this work, the consistent sampling requirement of signals is studied. We establish how this notion is related with certain set of projectors which are selfadjoint with respect to a semi-inner product. We extend previous results and present some new problems related with sampling theory.     

Local Sampling and Reconstruction in Shift-Invariant Spaces and Their Applications in Spline Subspaces

The local reconstruction from samples is one of the most desirable properties for many applications in signal processing. Local sampling is practically useful since we need only to consider a signal on a bounded interval and computer can only process finite samples. However, the local sampling and reconstruction problem has not been given as much attention. Most of known results concern global sampling and reconstruction. There are only a few results about local sampling and reconstruction in spline subspaces. In this article, we study local sampling and reconstruction in general shift-invariant spaces and multiple generated shift-invariant spaces with compactly supported generators. Then we give several applications in spline subspaces and multiple generated spline subspaces.     

Oblique Projections and Abstract Splines

Given a closed subspace of a Hilbert space and a bounded linear operator AL( ) which is positive, consider the set of all A-self-adjoint projections onto :

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In addition, if ₁ is another Hilbert space, T: → ₁ is a bounded linear operator such that T^*T=A and ξ , consider the set of (T, ) spline interpolants to ξ:

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A strong relationship exists between (A, ) and sp(T, ,ξ). In fact, (A, ) is not empty if and only if s p(T, ,ξ) is not empty for every ξ . In this case, for any ξ it holds

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and for any ξ , the unique vector of s p(T, ,ξ) with minimal norm is (1−P_A, )ξ, where P_A, is a distinguished element of (A, ). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators.     

Irregular Sampling in Wavelet Subspaces

As a particular wavelet subspace, the Paley-Wiener space has both regular and irregular sampling theorems. A regular sampling theorem in general wavelet subspaces has been established for several years. In this paper, we discuss the irregular sampling problem in wavelet subspaces.     

Positive Sampling in Wavelet Subspaces

This paper is devoted to the discussion of a “hybrid” sampling series, a series of translates of a nonnegative summability function used in place of an orthogonal scaling function. The coefficients in the series are taken to be sampled values of the function to be approximated. This enables one to avoid the integration which arises in the other series. The approximations based on this hybrid series have certain desirable convergence properties: they are locally uniformly convergent for locally continuous functions, they have quadratic uniform convergence rate for functions in certain Sobolev spaces, they are locally bounded when the function is locally bounded and therefore, in particular, Gibbs' phenomenon is avoided. Numerical experiments are given to illustrate the theoretical results and to compare these approximations with the scaling function approximations.     

Invariant Subspaces for Pairs of Projections

A simple geometrical argument shows that every pair of projectionson a finite-dimensional complex vector space has a common invariantsubspace of dimension 1 or 2. The idea extends to certain pairsof projections on an infinite-dimensional Hilbert space H. Inparticular every projection on H has a reducing subspace, althougha finite-dimensional one need not exist. In a final section,the results are extended to the existence of hyperinvariantsubspaces for pairs of projections.     

Minimal Projections onto Subspaces with Codimension 2

In this article, the relationship between absolute projection constants with dimension n and with codimension n is given. In particular, bounds from below are found for absolute projection constants with codimension 2. Also, exact formulas for minimal projections in this case are given. It is shown that these minimal projections are not unique.     

Weak Chebyshev Subspaces and Continuous Selections for Parametric Projections

We examine the existence of continuous selections for the parametric projection onto weak Chebyshev subspaces. In particular, we show that if is the class of polynomial splines of degree n with the k fixed knots then the parametric projection admits a continuous selection if and only if the number of knots does not exceed the degree of splines plus one. February 15, 1996. Date revised: September 16, 1996.     

主平移不变子空间的平移整采样

给出主平移不变子空间的一个平移整采样定理,其采样公式不仅在L2(R)收敛意义下成立,而且在适当的1周期集上一致收敛的意义下成立.此采样定理包含了经典的Shannon采样公式,Walter在1992年的采样定理以及由紧支函数生成的主平移不变子空间的采样.最后给出了大量例子说明定理应用的广泛性.     

Aliasing Error of Sampling Series in Wavelet Subspaces

Alising error arises whenever a sampling formula, valid for a prescribed space, is applied to a function in a bigger space. In this work, we estimate the aliasing error of classic and average sampling expansions in wavelet subspaces of a multiresolution analysis.     

Minimal Multi-Convex Projections Onto Subspaces of Incomplete Algebraic Polynomials

Let X = (C ^N [0, 1], ‖·‖), where N ≥ 3 and let V be a linear subspace of Π _N , where Π _N denotes the space of algebraic polynomials of degree less than or equal to N. Denote by 𝒫 _S = 𝒫 _S (X, V) = {P: X → V | P-linear and bounded P| _V  = id _V , PS ? S}, where S denotes a cone of multi-convex functions. In [25 G. Lewicki and M. Prophet ( 2006 ). Minimal shape-preserving projections onto Π _n : generalizations and extensions . Numer. Func. Anal. Optim. 27 ( 7–8 ): 847 – 873 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 26 G. Lewicki and M. Prophet ( 2007 ). Minimal multi-convex projections . Studia Mathematica 178 ( 2 ): 99 – 124 . [Google Scholar]], the multi-convex projections were defined and it was shown the explicite formula for projection with minimal norm in 𝒫 _S for V = Π _N . In this article we present a generalization of these results in the case of V being certain, proper subspaces of Π _N .     

Weighted Generalized Inverses,Oblique Projections,and Least-Squares Problems

A generalization with singular weights of Moore–Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.     

投影的约化极小模和空间的夹角

Let M and N be nonzero subspaces of a Hilbert space H, and PM and PN denote the orthogonal projections on M and N, respectively. In this note, an exact representation of the angle and the minimum gap of M and N is obtained. In addition, we study relations between the angle, the minimum gap of two subspaces M and N, and the reduced minimum modulus of （I - PN）PM,     

Sampling Formulas Involving Differences in Shift-Invariant Subspaces: A Unified Approach

Successive differences on a sequence of data help discover some smoothness features of this data. This was one of the main reasons for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper is to mimic them to a sequence of regular samples of a function in a shift-invariant subspace allowing its stable recovery. A suitable expression for the functions in the shift-invariant subspace by an isomorphism with the L²(0,1) space is the key to identify the simple pattern followed by the dual Riesz bases involved in the derived formulas. The paper contains examples illustrating different non-exhaustive situations including also the two-dimensional case.     

A Quadrature Formula Based On Sampling In Meyer Wavelet Subspaces

In this paper we discuss a weighted trapezoidal rule based on sampling in Meyer wavelet subspaces. For a wide class of functions, we obtain convergence and error bounds. Some examples are given to construct sampling functions.     

小波空间的非均匀采样及其重构

信号的采样问题，就是探讨采样集满足什么条件时，能够重建信号，如何重建信号．对于f(x)∈L^2(R)，这里证明了，当采样集满足一定的条件时，适当选择小波基，可以重建信号，并且考虑了用迭代重构算法来重建信号，得到了具体的逼近精度．     

Optimal Rates of Linear Convergence of Relaxed Alternating Projections and Generalized Douglas-Rachford Methods for Two Subspaces

We systematically study the optimal linear convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Our analysis is based on a study on the linear convergence rates of the powers of matrices. We show that the optimal linear convergence rate of powers of matrices is attained if and only if all subdominant eigenvalues of the matrix are semisimple. For the convenience of computation, a nonlinear approach to the partially relaxed alternating projection method with at least the same optimal convergence rate is also provided. Numerical experiments validate our convergence analysis     

A Block Sampling Approach to Distinct Value Estimation

To process queries efficiently in a relational database it is often of value to estimate the number of distinct values occurring in a particular field. In contexts where complete enumeration is costly, estimators based on a subsample are an attractive alternative. Though many such estimators have already been proposed, most are based on a simple random sampling of records which is wasteful in a database where records are retrieved in blocks. This article seeks to develop estimators that perform well in contexts where blocks of records are sampled randomly. Building on a result of Good's we derive six such estimators and evaluate their performance.