Radial limits of harmonic functions on the unit disc

This note characterizes those functions on the unit circle that can arise as the radial limit function of a harmonic function on the unit disc.

     

On the Hilbert formulas on the unit sphere for the time-harmonic relativistic Dirac bispinors theory

In this paper there are established some analogues of the Hilbert formulas on the unit sphere for the theory of time-harmonic (monochromatic) relativistic Dirac bispinors. The formulas relate a pair of the components of the limit value of a time-harmonic Dirac bispinor in the unit ball to the other pair of components. The obtained results are based on the intimate connection between time-harmonic solutions of the relativistic Dirac equation and the three-dimensional α-hyperholomorphic function theory. Hilbert formulas for α-hyperholomorphic function theory for α being a complex quaternionic number are also presented, such formulas relate a pair of components of the boundary value of an α-hyperholomorphic function in the unit ball to the other pair of components, in an analogy with what is known for the case of the theory of functions of one complex variable.     

Spherical analog of the complex form of trigonometric fourier series; Approximation

For a function defined on the unit sphere of dimension at least 2, we suggest a spherical analog with algebraic structure similar to the Fourier components of a function defined on the unit circle. The deviations of the analogs of partial Fourier sums from the function are estimated with respect to various norms. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 764–777, May, 2000.     

Spectral factorization in the disk algebra

Every strictly positive function f, given on the unit circle of the complex plane, defines an outer function. This article investigates the behavior of these outer functions on the boundary of the unit disk. It is shown that even if the given function f on the boundary is continuous, the corresponding outer function is generally not continuous on the closure of the unit disk. Moreover, any subset E∈ [-π ,π) of Lebesgue measure zero is a valid divergence set for outer functions of some continuous functions f. These results are applied to study the solutions of non-linear boundary-value problems and the factorization of spectral density functions.     

On the mean value property for polyharmonic functions in the ball

We obtain the mean value property for the normal derivatives of a polyharmonic function with respect to the unit sphere. We find the values of a polyharmonic function and its Laplacians at the center of the unit ball expressed via the integrals of the normal derivatives of this function over the unit sphere.     

Nontrivial discontinuities of the Golovach functions for trees

The ɛ-search problem on graphs is considered. Properties of the Golovach function, which associates each nonnegative number ɛ with the ɛ-search number, are studied. It is known that the Golovach function is piecewise constant, nonincreasing, and right continuous. Golovach and Petrov proved that the Golovach function for a complete graph on more than five vertices may have nonunit jumps. The jumps of the Golovach function for the case of trees are considered. Examples of trees which disprove the conjecture that the Golovach function has only unit jumps for any planar graph are given. For these examples, the Golovach function is constructed. It is shown that the Golovach function for trees with at most 27 edges has only unit jumps. The same assertion is proved for trees containing at most 28 edges all of whose vertices have degree at most 3. The examples mentioned above have minimum number of edges.     

The ¯∂-problem for multipliers of the Sobolev space

For , let and be the spaces of multipliers of , the Sobolev space on the unit circle, and , the Dirichlet type space on the open unit disk, respectively. In fact, and are obtained from and by analytic extension. In this paper, we show that if is an -Carleson measure on the open unit disk, then there exists a function f defined on the closed unit disk such that the equation holds on the open unit disk, and such that the boundary value function f belongs to . For applications, we first establish the corona theorem for , which, in the case , gives the answer to a question of L. Brown and A. L. Shields. Secondly, we obtain a geometric characterization of the interpolating sequences for with that extends a theorem of D. E. Marshall and C. Sundberg. Received: 20 October 1997 / Revised version: 7 May 1998     

Points of Weak-Norm Continuity in the Unit Ball of Banach Spaces

Using the M-structure theory, we show that several classical function spaces and spaces of operators on them fail to have points of weak-norm continuity for the identity map on the unit ball. This gives a unified approach to several results in the literature that establish the failure of strong geometric structure in the unit ball of classical function spaces. Spaces covered by our result include the Bloch spaces, dual of the Bergman space L¹_a and spaces of operators on them, as well as the space C(T)/A, where A is the disc algebra on the unit circle T. For any unit vector f in an infinite-dimensional function algebra A we explicitly construct a sequence {f_n} in the unit ball of A that converges weakly to f but not in the norm.     

Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball

For the standard weighted Bergman spaces on the complex unit ball, the Berezin transform of a bounded continuous function tends to this function pointwise as the weight parameter tends to infinity. We show that this remains valid also in the context of harmonic Bergman spaces on the real unit ball of any dimension. This generalizes the recent result of C. Liu for the unit disc, as well as the original assertion concerning the holomorphic case. Along the way, we also obtain a formula for the corresponding weighted harmonic Bergman kernels.

     

A property of Hermite-Padé interpolation on the roots of unity

We extend a theorem of Ivanov and Saff to show that for the Hermite-Padé interpolant at the roots of unity to a function meromorphic in the unit disc, its leading coefficients vanish if and only if the corresponding interpolant to a related function vanishes at given points outside the unit disc. The result is then extended to simultaneous Hermite-Padé inter polation to a finite set of functions.     

Stability of the Cournot equilibrium for a Cournot oligopoly model with n competitors

In this paper a Cournot-like model is constructed with an iso-elastic demand function for n competitors. The Cournot equilibrium is constructed for general constant unit costs. Finally, it is proved that for identical unit costs the Cournot point is a sink for two or three competitors and a saddle for more than four players.     

A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval

A method for construction of CF approximants in some cases of rational approximation of a rational function f on the unit disk and on the unit interval is presented. The inverted square root of the greatest positive eigenvalue and a corresponding eigenvector of an eigenvalue problem defined by the coefficients of f gives the solution.     

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.     

Set of ambiguous points for functions in ℝ n

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point ζ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint ζ, such that cluster sets of function along these arcs are disjoint. We investigate whether it is possible to modify the notion of ambiguous point to keep the analogous result true for functions defined in the n-dimensional Euclidean unit ball.     

Boundedness of the l-Index of the Naftalevich–Tsuji Product

We investigate conditions for zeros under which the Naftalevich–Tsuji product is a function of a bounded l-index analytic in the unit disk.     

A perfect example for the BFGS method

Consider the BFGS quasi-Newton method applied to a general non-convex function that has continuous second derivatives. This paper aims to construct a four-dimensional example such that the BFGS method need not converge. The example is perfect in the following sense: (a) All the stepsizes are exactly equal to one; the unit stepsize can also be accepted by various line searches including the Wolfe line search and the Arjimo line search; (b) The objective function is strongly convex along each search direction although it is not in itself. The unit stepsize is the unique minimizer of each line search function. Hence the example also applies to the global line search and the line search that always picks the first local minimizer; (c) The objective function is polynomial and hence is infinitely continuously differentiable. If relaxing the convexity requirement of the line search function; namely, (b) we are able to construct a relatively simple polynomial example.     

Determining radii of meromorphy via orthogonal polynomials on the unit circle

Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szeg functions.     

Sobolev Trace Theorem and the Dirichlet Problem in the Unit Disk

We give a direct and elementary proof for the trace theorem in L _p -based Sobolev spaces, when the domain is the unit disk. We also consider the Dirichlet boundary problem for the Laplace equation, where the boundary value is a function in the Besov space. The Poisson kernel enables us to solve this problem in the unit disk more easily than in a general domain.     

Discrepancy and the Error in Integration

 This paper is devoted to an estimation of the error of integration with respect to arbitrary unit measures μ and ν on only in terms of continuity or smoothness properties of the function f and the discrepancy . Here, stands for certain classes of (Borel-) test sets. The proofs are in part based on a continuous wavelet analysis of the integrated function by means of Haar-type wavelets.     

On the sharpness of Meier’s analogue of Fatou’s theorem

Meier’s topological analogue of Fatou’s theorem is shown to be sharp by exhibiting a bounded holomorphic function in the unit disk for which no point of a prescribed set of first category on the unit circle is a Meier point. Supported by the U. S. Army Research Office, Durham.