Gabor analysis over finite Abelian groups

The topic of this paper are (multi-window) Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time-frequency plane. Our generic approach covers simultaneously multi-dimensional signals as well as non-separable lattices. The main results reduce to well-known fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler-Raz or Tolimieri-Orr, Bastiaans and Van-Leest, among others. In our presentation a central role is given to spreading function of linear operators between finite-dimensional Hilbert spaces. Another relevant tool is a symplectic version of Poisson's summation formula over the finite time-frequency plane. It provides the Fundamental Identity of Gabor Analysis.In addition we highlight projective representations of the time-frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames.Comment: Revised version: two new sections added, many typos fixe

Model of Enterpreneurship and Social-cultural and Market Orientation of Small Business Owners in Poland

In the development of SMEs in Poland crucial meaning is legislation, steadily adapted to EU regulations, especially to the European Charter for Small Enterprises. Research conducted in Poland by many authors provide data for doing so, to confirm the hypothesis that among small businesses a vital role in shaping their work situation did not continue to play the market mechanisms and orientations, but mainly socio-cultural factors.W rozwoju MŚP w Polsce podstawowe znaczenie mają również uregulowania prawne, systematycznie dostosowywane do regulacji unijnych, zwłaszcza zaś do Europejskiej Karty Małych Przedsiębiorstw. Badania prowadzone w Polsce przez wielu autorów dostarczają danych ku temu, by potwierdzić tezę, że wśród drobnych przedsiębiorców decydującą rolę w kształtowaniu ich sytuacji pracy odgrywają nadal nie mechanizmy i orientacje rynkowe, ale przede wszystkim czynniki społeczno-kulturowe

Parties, promiscuity and politicisation: business-political networks in Poland

Research on post-communist political economy has begun to focus on the interface between business and politics. It is widely agreed that informal networks rather than business associations dominate this interface, but there has been very little systematic research in this area. The literature tends to assume that a politicised economy entails business-political networks that are structured by parties. Theoretically, this article distinguishes politicisation from party politicisation and argues that the two are unlikely to be found together in a post-communist context. Empirically, elite survey data and qualitative interviews are used to explore networks of businesspeople and politicians in Poland. Substantial evidence is found against the popular idea that Polish politicians have business clienteles clearly separated from each other according to party loyalties. Instead, it is argued that these politicians and businesspeople are promiscuous. Since there seems to be little that is unusual about the Polish case, this conclusion has theoretical, methodological, substantive and policy implications for other post-communist countries

Cornerstones of Sampling of Operator Theory

This paper reviews some results on the identifiability of classes of operators whose Kohn-Nirenberg symbols are band-limited (called band-limited operators), which we refer to as sampling of operators. We trace the motivation and history of the subject back to the original work of the third-named author in the late 1950s and early 1960s, and to the innovations in spread-spectrum communications that preceded that work. We give a brief overview of the NOMAC (Noise Modulation and Correlation) and Rake receivers, which were early implementations of spread-spectrum multi-path wireless communication systems. We examine in detail the original proof of the third-named author characterizing identifiability of channels in terms of the maximum time and Doppler spread of the channel, and do the same for the subsequent generalization of that work by Bello. The mathematical limitations inherent in the proofs of Bello and the third author are removed by using mathematical tools unavailable at the time. We survey more recent advances in sampling of operators and discuss the implications of the use of periodically-weighted delta-trains as identifiers for operator classes that satisfy Bello's criterion for identifiability, leading to new insights into the theory of finite-dimensional Gabor systems. We present novel results on operator sampling in higher dimensions, and review implications and generalizations of the results to stochastic operators, MIMO systems, and operators with unknown spreading domains

Approximation of integral operators using product-convolution expansions

International audienceWe consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called product-convolution expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices

Bahadur Representation for the Nonparametric M-Estimator Under alpha-mixing Dependence

Under the condition that the observations, which come from a high-dimensional population (X,Y), are strongly stationary and strongly-mixing, through using the local linear method, we investigate, in this paper, the strong Bahadur representation of the nonparametric M-estimator for the unknown function m(x)=arg minaIE(r(a,Y)|X=x), where the loss function r(a,y) is measurable. Furthermore, some related simulations are illustrated by using the cross validation method for both bivariate linear and bivariate nonlinear time series contaminated by heavy-tailed errors. The M-estimator is applied to a series of S&P 500 index futures andspot prices to compare its performance in practice with the usual squared-loss regression estimator

Smoothing Quantile Regressions

We propose to smooth the objective function, rather than only the indicator on the check function, in a linear quantile regression context. Not only does the resulting smoothed quantile regression estimator yield a lower mean squared error and a more accurate Bahadur-Kiefer representation than the standard estimator, but it is also asymptotically differentiable. We exploit the latter to propose a quantile density estimator that does not suffer from the curse of dimensionality. This means estimating the conditional density function without worrying about the dimension of the covariate vector. It also allows for two-stage efficient quantile regression estimation. Our asymptotic theory holds uniformly with respect to the bandwidth and quantile level. Finally, we propose a rule of thumb for choosing the smoothing bandwidth that should approximate well the optimal bandwidth. Simulations confirm that our smoothed quantile regression estimator indeed performs very well in finite samples