The Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates

The material presented so far naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. In this final chapter we make connections to abstract harmonic analysis and show how we can gain insight about frames via the theory for group representations. More precisely, we show how the orthogonality relations for square-integrable group representations lead to series expansions of the elements in the underlying Hilbert space; on a concrete level, this gives an alternative approach to Gabor systems and wavelet systems. Feichtinger and Gröchenig proved that the group-theoretic setup even allows us to obtain series expansions in a large scale of Banach spaces, a result which leads Gröchenig to define frames in Banach spaces. By removing some of the conditions we obtain p-frames, first studied separately by Aldroubi, Sun, and Tang.

It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual (multi-) wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples that demonstrate the extra flexibility in the extension to dual pairs of frames compared with the more popular extensions to tight frames.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

It is shown that there exists a frame wavelet with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θa transform in , and using Θa‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.

In signal processing, discrete convolutions are usually involved in fast calculating coefficients of time-frequency decompositions like wavelet and Gabor frames. Depending on the regularity of the mother analyzing functions, one wants to detect the right resolution in order to achieve good approximations of coefficients. Local–global conditions on functions in order to get the convergence rate of Riemann-type sums to their scalar products in L 2 are presented. Wiener amalgam spaces, in particular for the space-time domain and W(L 2,l 1) for the frequency domain, give natural norms in order to estimate errors. In particular, relations between the rate of convergence of these series to integrals by increasing resolution and the (minimal) required Besov regularity are presented by means of functional and harmonic analysis techniques.

A new approach to fuzzy wavelet system modeling

The continuity of the optimal value function of a parametric convex semi-infinite program is secured by a weak regularity condition that also implies the convergence of certain discretization methods for semi-infinite problems. Since each discretization level yields a parametric program, a sequence of optimal value functions occurs. The regularity condition implies that, with increasing refinement of the discretization, this sequence converges uniformly with respect to the parameter to the optimal value function corresponding to the original semi-infinite problem. Our result is applicable to the convergence analysis of numerical algorithms based on parametric programming, for example, rational approximation and computation of the eigenvalues of the Laplacian.

This article considers the adaptive bridge estimator for Cox proportional hazards model with a diverging number of parameters. Specifically, we assume that the number of covariates of the Cox model is not fixed, and use the adaptive bridge penalty method to select variables and estimate the parameters of the Cox model. We show that under some weak regularity conditions, our proposed method has oracle properties and can select non-zero coefficients with a probability converging to 1. We also present an algorithm based on the Minorization-Maximization method to solve the adaptive bridge estimation. In addition, simulation studies show that the proposed method is effective and the well-known primary biliary cirrhosis data is provided to illustrate the application of the proposed method in survival problems.

We compare the probabilistic properties of the non-Gaussian Ornstein-Uhlenbeck based stochastic volatility model of Barndorff-Nielsen and Shephard (2001) with those of the COGARCH process. The latter is a continuous time GARCH process introduced by the authors (2004). Many features are shown to be shared by both processes, but differences are pointed out as well. Furthermore, it is shown that the COGARCH process has Pareto like tails under weak regularity conditions.

In this paper, the authors consider the behavior on BMO($\mathbb R^n$) and Campanato spaces for the higher-dimensional Marcinkiewicz integral operator which is defined by where Ω is homogeneous of degree zero, has mean value zero and is integrable on the unit sphere. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO($\mathbb R^n$) or to a certain Campanato space, then [μΩ(f)]2 is either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness is also obtained. The corresponding Lusin area integral is also considered.

Let X , X 1 , X 2 , … X,{X_1},{X_2}, \ldots be i.i.d. random variables. It is shown that E | X | log + log + | X | > ∞ E\left | X \right |{\log ^ + }{\log ^ + }\left | X \right | > \infty is a sufficient condition for Riemann R 1 {R_1} -summability of { X n } \left \{ {{X_n}} \right \} to E X EX . Counterexamples are provided which indicate that the strongest possible necessary condition of moment type is E | X | > ∞ E\left | X \right | > \infty . However under weak regularity conditions on the tails of the distribution of X X the sufficient condition is also shown to be necessary.

The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To fix this drawback of thresholding estimation, we develop a positive-definite ℓ1-penalized covariance estimator for estimating sparse large covariance matrices. We derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties. Under weak regularity conditions, nonasymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulation and real applications.

The paper proposes new instrumental variables estimators for the slope parameters of a panel data model with classical endogeneity in which all the observables – including the instruments – may have a common factors structure. These estimators are shown to be consistent and asymptotically normal under weak regularity conditions. A small Monte Carlo simulation shows that these estimators compare favourably to existing estimators.

A theory of identification is developed for general stochastic model whose probability law is determined by finite number of parameters. It is shown under weak regularity conditions that local identifiability of the unknown parameter vector is equivalent to nonsingularity of the information matrix. The use of reduced-form parameters to establish identifiability is also analyzed. The general results are applied to the familiar problem of determining whether the coefficients of system of linear simultaneous equations are identifiable. THE IDENTIFICATION PROBLEM concerns the possibility of drawing inferences from observed samples to an underlying theoretical An important part of econometric theory involves the derivation of conditions under which given structure will be identifiable. The basic results for linear simultaneous equation systems under linear parameter constraints were given by Koopmans and Rubin [10] in 1950. Extensions to nonlinear systems and nonlinear constraints were made by Wald [15], Fisher [4, 5, 6], and others. A summary of these results can be found in Fisher's comprehensive study [7]. The identification problem has also been thoroughly analyzed in the context of the classical single-equation errors-in-variables model. The basic papers here are by Neyman [12] and Reiers0l [13]. Most of this previous work on the identification problem has emphasized the special features of the particular model being examined. This has tended to obscure the fact that the problem of structural identification is very general one. It is not restricted to simultaneous-equation or errors-in-variables models. As Koopmans and Reiers0l [9] emphasize, the identification problem is a general and fundamental problem arising, in many fields of inquiry, as concomitant of the scientific procedure that postulates the existence of structure. In their important paper Koopmans and Reiers0l define the basic characteristics of the general identification problem. In the present paper we shall, in the case of general parametric model, derive some identifiability criteria. These criteria include the standard rank conditions for linear models as special cases. Our approach is based in part on the information matrix of classical mathematical statistics. Since this matrix is measure of the amount of information about the unknown parameters available in the sample, it is not surprising that it should be related to identification. For lack of identification is simply the lack of sufficient information to distinguish between alternative structures. The following results make this relationship more precise.2