Basic Polynomial Inequalities on Intervals and Circular Arcs

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szeg?–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.     

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.     

Local Poincaré inequalities in non-negative curvature and finite dimension

In this paper, we derive a new set of Poincaré inequalities on the sphere, with respect to some Markov kernels parameterized by a point in the ball. When this point goes to the boundary, those Poincaré inequalities are shown to give the curvature-dimension inequality of the sphere, and when it is at the center they reduce to the usual Poincaré inequality. We then extend them to Riemannian manifolds, giving a sequence of inequalities which are equivalent to the curvature-dimension inequality, and interpolate between this inequality and the Poincaré inequality for the invariant measure. This inequality is optimal in the case of the spheres.     

Conformal barycenters and the Douady-Earle extension—a discrete dynamical approach

The Douady-Earle extension produces a homeomorphism of a disk from a homeomorphism of its bounding circle. It is based on a center of mass computation at the center and is extended to the disk by naturality. Quasiconformal homeomorphisms are the extensions of quasisymmetric ones. There are several approaches to the numerical computation of the extension defined by a redistribution of mass. The algorithms of Abikoff-Ye and Milnor turn out to be the same —even in the more general situation of nontrivial probability measures on the unit circle. The numerical computation uses measures with finite support; in that case, the iterator has rational square. We obtain an easy approach to the proof of the validity of the algorithm and to its calculation. New proofs and additional properties of the computation of the barycenter and the extension are also presented. Much of this work was done while the author was a Lady Davis Visiting Professor at the Technion— Israel Institute of Technology. I gratefully acknowledge the hospitality and support of the Faculty of Mathematics there.     

Constructive approximations of spherical functions

Approximation of functions on the sphere arises in almost all applications modeling data collected on the surface of the earth and for reconstruction of various processes in spherical coordinates. Constructive approximation of high dimensional spherical functions are useful for approximation of processes of several variables on a compact subset of a Euclidean space by mapping the data onto the unit sphere of a space having one higher dimension, avoiding the boundary effect of the set. Interpolation operators on the circle and periodic domains (based on a class of basis functions and data values) are essential for many high performance simulations. These operators are represented by analytical summation formulas that can be computed very efficiently using the fast Fourier transform. This work is concerned with construction of a similar class of interpolation and quasi-interpolation operators on the unit sphere in ℝ^q , for q = 3, 4, 5, · · ·, using a new class of basis functions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the zeros of certain self-reciprocal polynomials

We investigate the distribution of zeros around the unit circle of real self-reciprocal polynomials of even degrees with five terms whose absolute values of middle coefficients equal the sum of all other coefficients. Furthermore, it also give a new inequality and other Eneström-Kakeya types of results as by-products of this investigation.     

A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle

We prove a general Borg-type result for reflectionless unitaryCMV operators U associated with orthogonal polynomials on theunit circle. The spectrum of U is assumed to be a connectedarc on the unit circle. This extends a recent result of Simonin connection with a periodic CMV operator with spectrum thewhole unit circle. In the course of deriving the Borg-type result we also use exponentialHerglotz representations of Caratheodory functions to provean infinite sequence of trace formulas connected with the CMVoperator U.     

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in ${\mathbb C^n}We compute the Szeg? kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb Cⁿ{\mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.     

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.     

球面S~(d-1)上的广义Ul'yanov型不等式

本文对于单位球面上的经典连续模,给出了一个非常有用的广义Ul'yanov型不等式.该不等式在球面多项式逼近、球面嵌入理论以及球面上函数空间的插值理论等领域有着非常重要的应用.我们的证明基于球面调和多项式展开的新的估计,这些估计本身也具有独立的意义.     

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k+1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang.     

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.     

Quadrature formulas on the unit circle based on rational functions

Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.     

Maximal function and multiplier theorem for weighted space on the unit sphere

For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex.     

Dual Toeplitz operators on the sphere

Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in Cn . In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones.     

Strong and Weak Convergence of Rational Functions Orthogonal on the Circle

Rational functions orthogonal on the unit circle with prescribedpoles lying outside the unit circle are studied. We establisha relation between the orthogonal rational functions and theorthogonal polynomials with respect to varying measures. Usingthis relation, we extend the recent results of Bultheel, González-Vera,Hendriksen and Njåstad on the asymptotic behaviour oforthogonal rational functions.     

On the Degree of Approximation by Spherical Translations

The degree of approximation of spherical functions by the translations formed by a function defined on the unit sphere is dealt with. A kind of Jackson inequality is established under the condition that none of the L^2（S^q） norms of the orthogonal projection operators of the translated function are zeros. In the present paper we show that the spherical translations share the same degree of approximation as that of spherical harmonics.     

Logarithmic Sobolev inequalities and hypercontractive estimates on the circle

A logarithmic Sobolev inequality, analogous to Gross' inequality, is proved on the circle. From this inequality it follows that the Poisson and heat semigroups on the circle satisfy Nelson's hypercontractive estimates.     

Failure of Polynomial Approximation on Polynomially Convex Subsets of the Sphere

An example is presented of a compact polynomially convex subsetof the unit sphere in C³ on which the polynomials in the complexcoordinate functions are not dense in the continuous functions.Also presented is an example of a compact polynomially convexsmooth solid torus lying in the unit sphere in C⁵ with the samefailure of polynomial approximation.     

Properties of interpolatory quadrature with equidistant nodes on the unit circle

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.