Radial Symbols Research Articles

Incrétines et diabète, une petite révolution conceptuelle et thérapeutique (3) L’effet incrétine, un nouveau concept thérapeutique

The space of Herglotz wave functions in R2 consists of all the solutions of the Helmholtz equation that can be represented as the Fourier transform in R2 of a measure supported in the circle and with density in L2(S1). This space has a structure of a Hilbert space with reproducing kernel. The purpose of this article is to study Toeplitz operators with nonnegative radial symbols, defined on this space. We study the symbols defining bounded and compact Toeplitz operators as well as the Toeplitz operators belonging to the Schatten classes sp.

Renormalised Multiple Integrals of Symbols with Linear Constraints

Given a holomorphic regularisation procedure (e.g. Riesz or dimensional regularisation) on classical symbols, we define renormalised multiple integrals of radial classical symbols with linear constraints. To do so, we first prove the existence of meromorphic extensions of multiple integrals of holomorphic perturbations of radial symbols with linear constraints and then implement either generalised evaluators or a Birkhoff factorisation. Renormalised multiple integrals are covariant and factorise over independent sets of constraints.

Commutative Algebras of Toeplitz Operators on the Reinhardt Domains

Let $D$ be a bounded logarithmically convex complete Reinhardt domain in $\mathbb{C}^n$ centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the $C^*$-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on $|z_1|$, $|z_2|$, ..., $|z_n|$) is commutative. We show that the natural action of the $n$-dimensional torus $\mathbb{T}^n$ defines (on a certain open full measure subset of $D$) a foliation which carries a transverse Riemannian structure having distinguished geometric features. Its leaves are equidistant with respect to the Bergman metric, and the orthogonal complement to the tangent bundle of such leaves is integrable to a totally geodesic foliation. Furthermore, these two foliations are proved to be Lagrangian. We specify then the obtained results for the unit ball.

Dynamics of Properties of Toeplitz Operators with Radial Symbols

In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T (λ) acting on weighted Bergman spaces % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFaeFqdaqhaaWcbaGa % eq4UdWgabaGae8NmaidaaOGaaiikamrr1ngBPrwtHrhAYaqehuuDJX % wAKbstHrhAGq1DVbacgaGae43GWtKaaiykaaaa!513C! $$\mathcal{A}_\lambda ^2 (\mathbb{D})$$ over the unit disk % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFdcpraaa!41CD! $$\mathbb{D}$$ , in dependence on λ, and compare their limit behavior under % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey % OKH4Qaey4kaSIaeyOhIukaaa!3BE2! $$\lambda \to + \infty $$ with corresponding properties of the initial symbol a.

Commuting Toeplitz Operators on the Pluriharmonic Bergman Space

We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.

Compact radial operators on the harmonic Bergman space

We study the characterizing problem of the compactness of radial operators on the harmonic Bergman space. We show that under an oscillation condition, the compactness is equivalent to the boundary vanishing conditions of the certain Berezin transforms. As an application, we characterize compact Toeplitz operators with radial symbol on the harmonic Bergman space.

Bergman-Toeplitz operators: Radial component influence

We analyze the influence of the radial component of a symbol to spectral, compactness, and Fredholm properties of Toeplitz operators, acting on the Bergman space. We show that there existcompact Toeplitz operators whose (radial) symbols areunbounded near the unit circle\(\partial \mathbb{D}\). Studying this question we give several sufficient, and necessary conditions, as well as the corresponding examples. The essential spectra of Toeplitz operators with pure radial symbols have sufficiently rich structure, and even can be massive.

A trace formula for Weyl transforms with radial symbols

A trace formula for Weyl transforms with radial symbols

Mellin Transform, Monomial Symbols, and Commuting Toeplitz Operators

In this paper we give necessary and sufficient conditions for a symbol that produces a Toeplitz operator on the Bergman space that commutes with another such operator whose symbol is a monomial. The result is stated in terms of the Mellin transform of the symbol. As a corollary, we show that a Toeplitz operator with a radial symbol may only commute with another such operator with a radial symbol.

The Visual Separability of Plotting Symbols in Scatterplots

Abstract Which symbols should be used to represent different groups of data in the same scatterplot? Hypotheses are derived to predict which symbol pairs should lead to good separability, based on the contrast of the symbols' visual properties or “features.” In two experiments, experimental scatterplots were shown to subjects on a computer screen; the dependent variable was the decision time to judge which of the two presented symbols was the more frequent one. Analyses of the within-subject effects yielded the following results: (1) Important feature contrasts are brightness, number of line endings, and curvature. (2) Symbols that differ simultaneously in two feature dimensions may be more separable than symbols that differ only in either one. (3) The contrasts between circular symbols and radial line symbols like the plus sign or the asterisk are excellent. Practical applications of these findings are discussed, as well as their contribution to the theory of visual perception.