Canonical analysis based on scatter matrices.

In this paper, the influence functions and limiting distributions of the canonical correlations and coefficients based on affine equivariant scatter matrices are developed for elliptically symmetric distributions. General formulas for limiting variances and covariances of the canonical correlations and canonical vectors based on scatter matrices are obtained. Also the use of the so called shape matrices in canonical analysis is investigated. The scatter and shape matrices based on the affine equivariant Sign Covariance Matrix as well as the Tyler's shape matrix are considered in more detail. Their finite sample and limiting efficiencies are compared to those of the Minimum Covariance Determinant estimator and S-estimates through theoretical and simulation studies. The theory is illustrated by an example.Canonical correlations; Canonical variables; Canonical vectors; Covariance; Covariance determinant estimator; Determinant estimator; Distribution; Efficiency; Estimator; Functions; Influence function; Matrix; Scatter; Shape matrix; Sign covariance mix; Simulation; Studies; Theory; Tyler's estimate;

Boundedness of Toeplitz Operators in Bergman-Type Spaces

Peer reviewe

Weak BMO and Toeplitz operators on Bergman spaces

Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function f is an element of BWMO is boundedly oscillating, and the analogous result holds for f is an element of VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.Peer reviewe

Pathology of essential spectra of elliptic problems in periodic family of beads threaded by a spoke thinning at infinity

We construct "almost periodic'' unbounded domains, where a large class of elliptic spectral problems have essential spectra possessing peculiar structure: they consist of monotone, non-negative sequences of isolated points and thus have infinitely many gaps.Peer reviewe

Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools

We consider the linear water-wave problem in a periodic channel pi(h)subset of R-3, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough h is proven.Peer reviewe

Boundedness of Toeplitz Operators in Bergman-Type Spaces

Peer reviewe

Spectral theory of Toeplitz and Hankel operators on the Bergman space A1

The Fredholm properties of Toeplitz operators on the Bergman space A2 have been well-known for continuous symbols since the 1970s. We investigate the case p=1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on Ap that arise when we no longer have 1<p<∞; in particular bounded Toeplitz operators on A1 were characterized completely very recently but only for bounded symbols. We also consider compactness of Hankel operators on A1

Weak BMO and Toeplitz operators on Bergman spaces

Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function f is an element of BWMO is boundedly oscillating, and the analogous result holds for f is an element of VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.Peer reviewe