Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

The images of Hermite and Laguerre-Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterized. These are used to characterize the image of Schwartz class of rapidly decreasing functions f on R-n and C-n under these semigroups. The image of the space of tempered distributions is also considered and a Paley-Wiener theorem for the windowed (short-time) Fourier transform is proved.

Semigroups on Frechet spaces and equations with infinite delays

In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions.

Semigroups on Frechet spaces and equations with infinite delays

In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Spacelike pion form factor from analytic continuation and the onset of perturbative QCD

The factorization theorem for exclusive processes in perturbative QCD predicts the behavior of the pion electromagnetic form factor F(t) at asymptotic spacelike momenta t(= -Q(2)) < 0. We address the question of the onset energy using a suitable mathematical framework of analytic continuation, which uses as input the phase of the form factor below the first inelastic threshold, known with great precision through the Fermi-Watson theorem from pi pi elastic scattering, and the modulus measured from threshold up to 3 GeV by the BABAR Collaboration. The method leads to almost model-independent upper and lower bounds on the spacelike form factor. Further inclusion of the value of the charge radius and the experimental value at -2.45 GeV2 measured at JLab considerably increases the strength of the bounds in the region Q(2) less than or similar to 10 GeV2, excluding the onset of the asymptotic perturbative QCD regime for Q(2) < 7 GeV2. We also compare the bounds with available experimental data and with several theoretical models proposed for the low and intermediate spacelike region.

On the structure of analytic vectors for the Schrodinger representation

This article deals with the structure of analytic and entire vectors for the Schrodinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.

A Technique to Compute Smooth Amplitude, Phase, and Frequency Modulations From the Analytic Signal

Gabor's analytic signal (AS) is a unique complex signal corresponding to a real signal, but in general, it admits infinitely-many combinations of amplitude and frequency modulations (AM and FM, respectively). The standard approach is to enforce a non-negativity constraint on the AM, but this results in discontinuities in the corresponding phase modulation (PM), and hence, an FM with discontinuities particularly when the underlying AM-FM signal is over-modulated. In this letter, we analyze the phase discontinuities and propose a technique to compute smooth AM and FM from the AS, by relaxing the non-negativity constraint on the AM. The proposed technique is effective at handling over-modulated signals. We present simulation results to support the theoretical calculations.

Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

We examine the large-order behavior of a recently proposed renormalization-group-improved expansion of the Adler function in perturbative QCD, which sums in an analytically closed form the leading logarithms accessible from renormalization-group invariance. The expansion is first written as an effective series in powers of the one-loop coupling, and its leading singularities in the Borel plane are shown to be identical to those of the standard ``contour-improved'' expansion. Applying the technique of conformal mappings for the analytic continuation in the Borel plane, we define a class of improved expansions, which implement both the renormalization-group invariance and the knowledge about the large-order behavior of the series. Detailed numerical studies of specific models for the Adler function indicate that the new expansions have remarkable convergence properties up to high orders. Using these expansions for the determination of the strong coupling from the hadronic width of the tau lepton we obtain, with a conservative estimate of the uncertainty due to the nonperturbative corrections, alpha(s)(M-tau(2)) = 0.3189(-0.0151)(+0.0173), which translates to alpha(s)(M-Z(2)) = 0.1184(-0.0018)(+0.0021). DOI: 10.1103/PhysRevD.87.014008

On Computing Amplitude, Phase, and Frequency Modulations Using a Vector Interpretation of the Analytic Signal

The amplitude-modulation (AM) and phase-modulation (PM) of an amplitude-modulated frequency-modulated (AM-FM) signal are defined as the modulus and phase angle, respectively, of the analytic signal (AS). The FM is defined as the derivative of the PM. However, this standard definition results in a PM with jump discontinuities in cases when the AM index exceeds unity, resulting in an FM that contains impulses. We propose a new approach to define smooth AM, PM, and FM for the AS, where the PM is computed as the solution to an optimization problem based on a vector interpretation of the AS. Our approach is directly linked to the fractional Hilbert transform (FrHT) and leads to an eigenvalue problem. The resulting PM and AM are shown to be smooth, and in particular, the AM turns out to be bipolar. We show an equivalence of the eigenvalue formulation to the square of the AS, and arrive at a simple method to compute the smooth PM. Some examples on synthesized and real signals are provided to validate the theoretical calculations.

Fractional Hilbert transform extensions and associated analytic signal construction

The analytic signal (AS) was proposed by Gabor as a complex signal corresponding to a given real signal. The AS has a one-sided spectrum and gives rise to meaningful spectral averages. The Hilbert transform (HT) is a key component in Gabor's AS construction. We generalize the construction methodology by employing the fractional Hilbert transform (FrHT), without going through the standard fractional Fourier transform (FrFT) route. We discuss some properties of the fractional Hilbert operator and show how decomposition of the operator in terms of the identity and the standard Hilbert operators enables the construction of a family of analytic signals. We show that these analytic signals also satisfy Bedrosian-type properties and that their time-frequency localization properties are unaltered. We also propose a generalized-phase AS (GPAS) using a generalized-phase Hilbert transform (GPHT). We show that the GPHT shares many properties of the FrHT, in particular, selective highlighting of singularities, and a connection with Lie groups. We also investigate the duality between analyticity and causality concepts to arrive at a representation of causal signals in terms of the FrHT and GPHT. On the application front, we develop a secure multi-key single-sideband (SSB) modulation scheme and analyze its performance in noise and sensitivity to security key perturbations. (C) 2013 Elsevier B.V. All rights reserved.

The Ginibre Ensemble and Gaussian Analytic Functions

We show that as n changes, the characteristic polynomial of the n x n random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to Polya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This suggests another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.

Binaural Signal Processing Motivated Generalized Analytic Signal Construction and AM-FM Demodulation

Binaural hearing studies show that the auditory system uses the phase-difference information in the auditory stimuli for localization of a sound source. Motivated by this finding, we present a method for demodulation of amplitude-modulated-frequency-modulated (AM-FM) signals using a ignal and its arbitrary phase-shifted version. The demodulation is achieved using two allpass filters, whose impulse responses are related through the fractional Hilbert transform (FrHT). The allpass filters are obtained by cosine-modulation of a zero-phase flat-top prototype halfband lowpass filter. The outputs of the filters are combined to construct an analytic signal (AS) from which the AM and FM are estimated. We show that, under certain assumptions on the signal and the filter structures, the AM and FM can be obtained exactly. The AM-FM calculations are based on the quasi-eigenfunction approximation. We then extend the concept to the demodulation of multicomponent signals using uniform and non-uniform cosine-modulated filterbank (FB) structures consisting of flat bandpass filters, including the uniform cosine-modulated, equivalent rectangular bandwidth (ERB), and constant-Q filterbanks. We validate the theoretical calculations by considering application on synthesized AM-FM signals and compare the performance in presence of noise with three other multiband demodulation techniques, namely, the Teager-energy-based approach, the Gabor's AS approach, and the linear transduction filter approach. We also show demodulation results for real signals.

Multiobjective fuzzy optimization for sustainable groundwater management using particle swarm optimization and analytic element method

Groundwater management involves conflicting objectives as maximization of discharge contradicts the criteria of minimum pumping cost and minimum piping cost. In addition, available data contains uncertainties such as market fluctuations, variations in water levels of wells and variations of ground water policies. A fuzzy model is to be evolved to tackle the uncertainties, and a multiobjective optimization is to be conducted to simultaneously satisfy the contradicting objectives. Towards this end, a multiobjective fuzzy optimization model is evolved. To get at the upper and lower bounds of the individual objectives, particle Swarm optimization (PSO) is adopted. The analytic element method (AEM) is employed to obtain the operating potentio metric head. In this study, a multiobjective fuzzy optimization model considering three conflicting objectives is developed using PSO and AEM methods for obtaining a sustainable groundwater management policy. The developed model is applied to a case study, and it is demonstrated that the compromise solution satisfies all the objectives with adequate levels of satisfaction. Sensitivity analysis is carried out by varying the parameters, and it is shown that the effect of any such variation is quite significant. Copyright (c) 2015 John Wiley & Sons, Ltd.