A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

Compression of Weighted Graphs

We propose to compress weighted graphs (networks), motivated by the observation that large networks of social, biological, or other relations can be complex to handle and visualize. In the process also known as graph simplication, nodes and (unweighted) edges are grouped to supernodes and superedges, respectively, to obtain a smaller graph. We propose models and algorithms for weighted graphs. The interpretation (i.e. decompression) of a compressed, weighted graph is that a pair of original nodes is connected by an edge if their supernodes are connected by one, and that the weight of an edge is approximated to be the weight of the superedge. The compression problem now consists of choosing supernodes, superedges, and superedge weights so that the approximation error is minimized while the amount of compression is maximized. In this paper, we formulate this task as the 'simple weighted graph compression problem'. We then propose a much wider class of tasks under the name of 'generalized weighted graph compression problem'. The generalized task extends the optimization to preserve longer-range connectivities between nodes, not just individual edge weights. We study the properties of these problems and propose a range of algorithms to solve them, with dierent balances between complexity and quality of the result. We evaluate the problems and algorithms experimentally on real networks. The results indicate that weighted graphs can be compressed efficiently with relatively little compression error.

Analogues of the Wiener-Tauberian and Schwartz theorems for radial functions on symmetric spaces

We prove a Wiener Tauberian theorem for the L-1 spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz-type theorem for complex groups. As a corollary we obtain a Wiener Tauberian type result for compactly supported distributions.

Part-of-Speech Tagging using Parallel Weighted Finite-State Transducers

We use parallel weighted finite-state transducers to implement a part-of-speech tagger, which obtains state-of-the-art accuracy when used to tag the Europarl corpora for Finnish, Swedish and English. Our system consists of a weighted lexicon and a guesser combined with a bigram model factored into two weighted transducers. We use both lemmas and tag sequences in the bigram model, which guarantees reliable bigram estimates.

Finite-State Spell-Checking with Weighted Language and Error Models

In this paper we present simple methods for construction and evaluation of finite-state spell-checking tools using an existing finite-state lexical automaton, freely available finite-state tools and Internet corpora acquired from projects such as Wikipedia. As an example, we use a freely available open-source implementation of Finnish morphology, made with traditional finite-state morphology tools, and demonstrate rapid building of Northern Sámi and English spell checkers from tools and resources available from the Internet.

BER Analysis of Weighted Interference Cancellation in Multicarrier DS-CDMA Systems

In this paper, we present the design and bit error performance analysis of weighted linear parallel interference cancellers (LPIC) for multicarrier (MC) DS-CDMA systems. We propose an LPIC scheme where we estimate (and cancel) the multiple access interference (MAI) based on the soft outputs on individual subcarriers, and the interference cancelled outputs on different subcarriers are combined to form the final decision statistic. We scale the MAI estimate on individual subcarriers by a weight before cancellation; these weights are so chosen to maximize the signal-to-interference ratios at the individual subcarrier outputs. For this weighted LPIC scheme, using an approach involving the characteristic function of the decision variable, we derive exact bit error rate (BER) expressions for different cancellation stages. Using the same approach, we also derive exact BER expressions for the matched filter (MF) and decorrelating detectors for the considered MC DS-CDMA system. We show that the proposed weighted LPIC scheme performs better than the MF detector and the conventional LPIC (where the weights are taken to be unity), and close to the decorrelating detector.

Weighted subspace methods and spatial smoothing: analysis and comparison

The effect of using a spatially smoothed forward-backward covariance matrix on the performance of weighted eigen-based state space methods/ESPRIT, and weighted MUSIC for direction-of-arrival (DOA) estimation is analyzed. Expressions for the mean-squared error in the estimates of the signal zeros and the DOA estimates, along with some general properties of the estimates and optimal weighting matrices, are derived. A key result is that optimally weighted MUSIC and weighted state-space methods/ESPRIT have identical asymptotic performance. Moreover, by properly choosing the number of subarrays, the performance of unweighted state space methods can be significantly improved. It is also shown that the mean-squared error in the DOA estimates is independent of the exact distribution of the source amplitudes. This results in a unified framework for dealing with DOA estimation using a uniformly spaced linear sensor array and the time series frequency estimation problems.

Toeplitz Operators with Special Symbols on Segal-Bargmann Spaces

We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.

Three- and Seven-Point Optimally Weighted Recursive Median Filters for Gas Turbine Diagnostics

Measured health signals incorporate significant details about any malfunction in a gas turbine. The attenuation of noise and removal of outliers from these health signals while preserving important features is an important problem in gas turbine diagnostics. The measured health signals are a time series of sensor measurements such as the low rotor speed, high rotor speed, fuel flow, and exhaust gas temperature in a gas turbine. In this article, a comparative study is done by varying the window length of acausal and unsymmetrical weighted recursive median filters and numerical results for error minimization are obtained. It is found that optimal filters exist, which can be used for engines where data are available slowly (three-point filter) and rapidly (seven-point filter). These smoothing filters are proposed as preprocessors of measurement delta signals before subjecting them to fault detection and isolation algorithms.

Off-line signature verification using genetically optimized weighted features

This paper is concerned with off-line signature verification. Four different types of pattern representation schemes have been implemented, viz., geometric features, moment-based representations, envelope characteristics and tree-structured Wavelet features. The individual feature components in a representation are weighed by their pattern characterization capability using Genetic Algorithms. The conclusions of the four subsystems teach depending on a representation scheme) are combined to form a final decision on the validity of signature. Threshold-based classifiers (including the traditional confidence-interval classifier), neighbourhood classifiers and their combinations were studied. Benefits of using forged signatures for training purposes have been assessed. Experimental results show that combination of the Feature-based classifiers increases verification accuracy. (C) 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.

Noise and outlier removal from jet engine health signals using weighted FIR median hybrid filters

The removal of noise and outliers from measurement signals is a major problem in jet engine health monitoring. Topical measurement signals found in most jet engines include low rotor speed, high rotor speed. fuel flow and exhaust gas temperature. Deviations in these measurements from a baseline 'good' engine are often called measurement deltas and the health signals used for fault detection, isolation, trending and data mining. Linear filters such as the FIR moving average filter and IIR exponential average filter are used in the industry to remove noise and outliers from the jet engine measurement deltas. However, the use of linear filters can lead to loss of critical features in the signal that can contain information about maintenance and repair events that could be used by fault isolation algorithms to determine engine condition or by data mining algorithms to learn valuable patterns in the data, Non-linear filters such as the median and weighted median hybrid filters offer the opportunity to remove noise and gross outliers from signals while preserving features. In this study. a comparison of traditional linear filters popular in the jet engine industry is made with the median filter and the subfilter weighted FIR median hybrid (SWFMH) filter. Results using simulated data with implanted faults shows that the SWFMH filter results in a noise reduction of over 60 per cent compared to only 20 per cent for FIR filters and 30 per cent for IIR filters. Preprocessing jet engine health signals using the SWFMH filter would greatly improve the accuracy of diagnostic systems. (C) 2002 Published by Elsevier Science Ltd.

On the explicit determination of the polar decomposition in n-dimensional vector spaces

A method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is proposed. The method uses the spectral resolution, and avoids the determination of eigenvectors when the tensor is invertible. For any given dimension n, an appropriately constructed van der Monde matrix is shown to play a key role in the construction of each of the component matrices (and their inverses) in the polar decomposition.

Probing the Allosteric Mechanism in Pyrrolysyl-tRNA Synthetase Using Energy-Weighted Network Formalism

Pyrrolysyl-tRNA synthetase (PyIRS) is an atypical enzyme responsible for charging tRNA(Pyl) with pyrrolysine, despite lacking precise tRNA anticodon recognition. This dimeric protein exhibits allosteric regulation of function, like any other tRNA synthetases. In this study we examine the paths of allosteric communication at the atomic level, through energy-weighted networks of Desulfitobacterium hafniense PyIRS (DhPyIRS) and its complexes with tRNA(Pyl) and activated pyrrolysine. We performed molecular dynamics simulations of the structures of these complexes to obtain an ensemble conformation-population perspective. Weighted graph parameters relevant to identifying key players and ties in the context of social networks such as edge/node betweenness, closeness index, and the concept of funneling are explored in identifying key residues and interactions leading to shortest paths of communication in the structure networks of DhPylRS. Further, the changes in the status of important residues and connections and the costs of communication due to ligand induced perturbations are evaluated. The optimal, suboptimal, and preexisting paths are also investigated. Many of these parameters have exhibited an enhanced asymmetry between the two subunits of the dimeric protein, especially in the pretransfer complex, leading us to conclude that encoding of function goes beyond the sequence/structure of proteins. The local and global perturbations mediated by appropriate ligands and their influence on the equilibrium ensemble of conformations also have a significant role to play in the functioning of proteins. Taking a comprehensive view of these observations, we propose that the origin of many functional aspects (allostery rand half-sites reactivity in the case of DhPyIRS) lies in subtle rearrangements of interactions and dynamics at a global level.