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Prolate spheroidal wavelets were previously introduced and shown to have some interesting convergence properties. In this work, several shortcomings of standard wavelets are discussed, and are shown not to be present in these new wavelets. These include invariance under arbitrary translations and differentiation of the associated multiresolution subspaces as well as similar properties of dilations.

The article is concerned with a particular multiresolution analysis (MRA) composed of Paley–Wiener spaces. Their usual wavelet basis consisting of sinc functions is replaced by one based on prolate spheroidal wave functions (PSWFs) which have much better time localization than the sinc function. The new wavelets preserve the high energy concentration in both the time and frequency domain inherited from PSWFs. Since the size of the energy concentration interval of PSWFs is one of the most important parameters in some applications, we modify the wavelets at different scales to retain a constant energy concentration interval. This requires a slight modification of the dilation relations, but leads to locally positive kernels. Convergence and other related properties, such as Gibbs phenomenon, of the associated approximations are discussed. A computationally friendly sampling technique is exploited to calculate the expansion coefficients. Several numerical examples are provided to illustrate the theory.

The prolate spheroidal wave functions (PSWFs) are used in sampling of bandlimited signals. Several formulae based on integer values of these PSWFs are derived and used to replace the sinc function in sampling theorems. They are also used to construct analysis and synthesis filter banks for sampled values. A type of multiresolution analysis based on PSWFs similar to that of wavelet theory is shown to exist. Several numerical examples illustrating the theory are presented.

Gibbs phenomenon occurs for most approximations based on standard orthogonal expansions, as well as for those based on integral operators. It also occurs in interpolations and other types of approximations. We consider a general approach to approximation based on delta sequences in an attempt to better understand the concept.

A method of calculating indices of annual year class strength from abundance data only is introduced. It involves a mathematical technique based on difference equations. The method is applied to data for six commercially harvested species of Green Bay, Lake Michigan. Estimated year class strengths are then compared with relative abundance indices of the parental spawning populations. Only for smelt and perch was there a significant correlation between the two. The same species were also the only ones for which the exponential relation of Ricker between spawners and year class strengths could be established.

Abstract: In this paper we prove general sampling theorems for functions belonging to a reproducing kernel Hilbert space (RKHS) which is also a closed subspace of a particular Sobolev space. We present details of this approach as applied to the standard sampling theory and its extension to nonuniform sampling. The general theory for orthogonal sampling sequences and nonorthogonal sampling sequences is developed. Our approach includes as concrete cases many recent extensions, for example, those based on the Sturm-Liouville transforms, Jacobi transforms, Laguerre transforms, Hankel transforms, prolate spherical transforms, etc., finite-order sampling theorems, as well as new sampling theorems obtained by specific choices of the RKHS. In particular, our setting includes nonorthogonal sampling sequences based on the theory of frames. The setting and approach enable us to consider various types of errors (truncation, aliasing, jitter, and amplitude error) in the same general context.

A multiresolution analysis for an orthogonal family of wavelets is usually not translation invariant. A concept of weak translation invariance is introduced and shown to hold for a class of Meyer wavelets and in fact characterizes this class. Other operators such as dilation, differentiation, and convolution are shown to have similar invariance properties for the same class.Copyright 1994, 1999 Academic Press

Periodic wavelets can be constructed from most standard wavelets by periodization. In this work we first derive some of their properties and then construct the periodic wavelets directly from their Fourier series without reference to standard wavelets. Several examples are given some of which are not constructable from the usual wavelets on the real line.

The impulse train obtained by sampling an analog signal is studied directly. It is shown to represent the signal in a weak sense and to converge to the signal as the sampling interval converges to 0. The convergence is in the sense of Sobolev spaces and in the sense of another Hilbert space for which the shifted “delta functions” are orthogonal.

A class of spectral windows depending on one parameter is presented and shown to include many of the common windows. The mean square rate of convergence of the associated spectral density estimators are calculated in terms of this parameter for spectral densities which are locally Lipschitz continuous The class is shown to include certain data tapers and data windows corresponding to missing observations. This is true also for the kernels of (C−α) summability which provide means for estimating the spectral density when the covariance function is periodic.

The basic splines are not orthogonal in L2and hence wavelets based on them are either not orthogonal or do not have compact support. However by choosing an appropriate norm in a Sobolev space, they and the wavelets based on them can be made orthogonal. Thus the simplicity and elegance of the original basic spline functions can be retained.

Abstract: Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.

Compartmental models, by which flows through various systems can be studied, have a dual aspect: one structural and the other dynamic. The structural leads to a directed graph and may be analyzed by means of graph theory. The dynamic leads to a system of differential equations which are usually linear. The coefficient matrix in this case has a special form, that of the negative transpose of an M-matrix in which the off diagonal elements are nonnegative and the columns add up to zero. Hence the eigenvalues have a nonpositive real part.If the digraph is weakly connected, the differential equation has a stable equilibrium solution; if it is unilaterally connected, the solution is unique; if it is strongly connected, the solution is feasible as well. It is possible to define various indices of stability which may then be shown to be related to indices of complexity of the structure. However, it is also possible, by redirecting the flows, to show that a given model can be reduced to a mammillary system with the same equilibrium solution. Hence any index of stability based on the equilibrium solution has no relation to a complexity index based on the number of arcs per vertex. Other stability indices, however, increase with increasing complexity.

This paper investigates the relationship between the analyticity of a function [image omitted]  and its wavelet coefficients [image omitted]  where [image omitted] , with [image omitted]  and ϕ is a Meyer-type mother wavelet. It is shown that if f is analytic in a neighborhood of the real axis, then the wavelet coefficients must satisfy certain growth conditions. Conversely, a sufficient condition on the coefficients is given to ensure that f is analytic in a neighborhood of the real line. Another sufficient condition on the coefficients is given which guarantees that f is an entire function.

The analytic representation of a function or a distribution on the real line is given in terms of series of "analytic wavelets." These series converge uniformly in compact subsets of the upper and lower half plane. The analytic wavelets used satisfy the same dilation equation as the orthogonal wavelets on the real axis.

The differential operator associated with the prolate spheroidal functions commutes with a certain integral operator. Its kernel is the reproducing kernel of the Paley-Wiener space restricted to a bounded interval. This "lucky accident" occurs because both the differential operator and its Fourier transform commute with multiplication by the characteristic function of an interval. It is the only operator of its type to do so.

The paper deals with a generalization of the projection pursuit regression algorithm (see P.Huber [4]) in Hilbert space. Theorem 2.1 states the strong convergence of the generalized algorithm. This result then applied to certain spaces in order to define a new density estimator and to obtain results on density approximation and estimation

The analytic properties of continuous Legendre transform $ F(\lambda )$ holomorphic in an elliptical neighborhood of the real interval $ [ - 1,1]$ has a singularity at $ {z_0}$ has one at $ {t_0}$ $ {z_0} = \cosh {t_0}$ in terms of .

It was shown by Janssen in 1993 that there are no continuous non-negative orthogonal scaling functions. For density estimation, we would like the estimators themselves to be densities, and hence non-negative. Thus the usual linear projection or the threshold estimators won't work without modification. In this paper, we introduce smooth non-negative wavelet estimators. The density estimators based on them are themselves density functions. These are applied to two numerical examples and compared to the usual wavelet estimators in these examples. An associated noise reduction method is considered.

This paper is concerned with the problem of estimating a generalized density of type [image omitted]  (δ denotes the Dirac delta) corresponding to a probability distribution function without a singular part. We propose two natural classes of estimates and obtain consistency results with respect to suitable norms.Our approach could be useful to provide an alternative model for those problems (simulation, smoothed bootstrap) in which density estimates are used as auxiliary tools to draw artificial samples. In particular, we suggest the possibility of considering a sort of mixed bootstrap, placed in an intermediate position between the two usual versions (smoothed and unsmoothed) of this technique.Certain interesting sequences of integer-valued random variables [called rare numbers by E. Key (1991)] arise, in a natural way, in the consistency proofs.