On the hardy space H¹ on products of half-spaces

We show that the Hardy space H¹ anal (R2+ x R2+) can be identified with the class of functions f such that f and all its double and partial Hubert transforms Hk f belong to L¹ (R2). A basic tool used in the proof is the bisubharmonicity of |F|q, where F is a vector field that satisfies a generalized conjugate system of Cauchy-Riemann type.

Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly- Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane.

The resonance spectrum of the cusp map in the space of analytic functions

We prove that the Frobenius-Perron operator $U$ of the cusp map $F:[-1,1]\to [-1,1]$, $F(x)=1-2 x^{1/2}$ (which is an approximation of the Poincare section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that for any $q\in (0,1)$ the spectrum of $U$ in the Hardy space in the disk $\{z\in C:|z-q|

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.

A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols

We characterize the essential spectra of Toeplitz operators Ta on weighted Bergman spaces with matrix-valued symbols; in particular we deal with two classes of symbols, the Douglas algebra C+H∞ and the Zhu class Q := L∞ ∩VMO∂ . In addition, for symbols in C+H∞ , we derive a formula for the index of Ta in terms of its symbol a in the scalar-valued case, while in the matrix-valued case we indicate that the standard reduction to the scalar-valued case fails to work analogously to the Hardy space case. Mathematics subject classification (2010): 47B35,

Hankel and Toeplitz transforms on H 1: continuity, compactness and Fredholm properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator H a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H a to be compact on H 1. The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2.

Tridiagonal Reproducing Kernels and Subnormality

We consider analytic reproducing kernel Hilbert spaces H with orthonormal bases of the form {(a(n) + b(n)z)z(n) : n >= 0}. If b(n) = 0 for all n, then H is a diagonal space and multiplication by z, M-z, is a weighted shift. Our focus is on providing extensive classes of examples for which M-z is a bounded subnormal operator on a tridiagonal space H where b(n) not equal 0. The Aronszajn sum of H and (1 - z)H where H is either the Hardy space or the Bergman space on the disk are two such examples.

A Note on Div-Curl Lemma

2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.

Hardy-Weinberg Equilibrium and the Ternary Plot

The Hardy-Weinberg law, formulated about 100 years ago, states that under certain assumptions, the three genotypes AA, AB and BB at a bi-allelic locus are expected to occur in the proportions p2, 2pq, and q2 respectively, where p is the allele frequency of A, and q = 1-p. There are many statistical tests being used to check whether empirical marker data obeys the Hardy-Weinberg principle. Among these are the classical xi-square test (with or without continuity correction), the likelihood ratio test, Fisher's Exact test, and exact tests in combination with Monte Carlo and Markov Chain algorithms. Tests for Hardy-Weinberg equilibrium (HWE) are numerical in nature, requiring the computation of a test statistic and a p-value. There is however, ample space for the use of graphics in HWE tests, in particular for the ternary plot. Nowadays, many genetical studies are using genetical markers known as Single Nucleotide Polymorphisms (SNPs). SNP data comes in the form of counts, but from the counts one typically computes genotype frequencies and allele frequencies. These frequencies satisfy the unit-sum constraint, and their analysis therefore falls within the realm of compositional data analysis (Aitchison, 1986). SNPs are usually bi-allelic, which implies that the genotype frequencies can be adequately represented in a ternary plot. Compositions that are in exact HWE describe a parabola in the ternary plot. Compositions for which HWE cannot be rejected in a statistical test are typically “close" to the parabola, whereas compositions that differ significantly from HWE are “far". By rewriting the statistics used to test for HWE in terms of heterozygote frequencies, acceptance regions for HWE can be obtained that can be depicted in the ternary plot. This way, compositions can be tested for HWE purely on the basis of their position in the ternary plot (Graffelman & Morales, 2008). This leads to nice graphical representations where large numbers of SNPs can be tested for HWE in a single graph. Several examples of graphical tests for HWE (implemented in R software), will be shown, using SNP data from different human populations

Best Constant in the Weighted Hardy Inequality: The Spatial and Spherical Version

Mathematics Subject Classification: 26D10.

Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas

2010 Mathematics Subject Classification: Primary 65D30, 32A35, Secondary 41A55.