Non-Essential Cluster Values and Normal Functions

Let D be the open unit disk in the complex plane, and let C be its boundary, the unit circle.If x £ C, then by an arc at x we mean a simple arc y with one end point at x such that y-{x}£ö.If/is a function mapping D into some metric space M, then the set of curvilinear convergence of/is defined to be {x e C: there exists an arc y at x and there exists a point p e M such that f(z) ->p as z -»■ x along y}.If ^ is a function whose domain is a subset F of the set of curvilinear convergence of/ then <f> is called a boundary function for/if, and only if, for each x e E there exists an arc y at x such that/(z) -*■ <f>(x) as z -> x along y.Let 5 be another metric space.We shall say that a function <f> is of Baire class ^ 1(5, M) if (i) domain <£ = S, (ii) range ^ £ M, and(iii) there exists a sequence {<£"} of continuous functions, each mapping S into M, such that <f>n^>-<p pointwise on 5. We shall say that (f> is of honorary Baire class ^ 2(S, M) if (i) domain <f> = S, (ii) range ^£M, and (iii) there exists a countable set TVs 5 and there exists a function y!r of Baire class ^ 1(5, M) such that <f>(x) = tp(x) for every xe S-N.It is known that if/is a continuous function mapping F into the Riemann sphere, then the set of curvilinear convergence off is of type Fa6, and any boundary function for/is of honorary Baire class ^2(C, Riemann sphere).(See [3], [4], [5], [6],[9].) J. E. McMillan [6] posed the following problem.If A is a given set in C of type Fao, and if </> is a function of honorary Baire class Ú2(A, Riemann sphere), does there always exist a continuous function / mapping D into the Riemann sphere such that A is the set of curvilinear convergence of/ and <f> is a boundary function for /?The purpose of this paper is to give an affirmative answer to McMillan's question.However, the corresponding question for real-valued functions remains open.(See Problems 1 and 2 at the end of this paper.)In proving our result, we first give a proof under the assumption that <j> is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere.As we shall indicate in an appendix, the same device can be

A known result of Newman and Tse asserts that every uniformly discrete sequence contained in a Stolz angle is uniformly separated (see Newman, D.J., 1959, Interpolation in . Transactions of the American Mathematical Society, 92(3), 501–507; Tse, K.-F., 1971, Nontangential interpolating sequences and interpolation by normal functions. Proceedings of the American Mathematical Society, 29, 351–354). We prove that this statement no longer holds if the sequence is located in a tangential region of certain kind. It is well known that a uniformly discrete sequence need not be a Blaschke sequence. We show, however, that every uniformly discrete sequence inside a disc tangential to the unit circle must be a Blaschke sequence.

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.

Let C and D be the unit circle and the open unit disk respectively. We shall use p(z,z′) to represent the non-Euclidean distance [3, p. 263] between the two points z and z′ in D, and X(w, w′) to represent the chordal distance between the two points w and w′ on the Riemann Sphere Ω.

Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane

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.

On the Polynomials Orthogonal on Regular Polygons

On fully operator Lipschitz functions

Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

A T T -symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ [ T ] ( ζ ) = [ ϕ ( ζ T ) ] 1 ∕ T {\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ \zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

We investigate the growth rate of the Birkhoff sums $S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb{T}$ and $(\unicode[STIX]{x1D6FC},x)$ is a ‘typical’ element of $\mathbb{T}^{2}$. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.

An integral operator induced by a Zygmund function

In this article we calculate the exact location in the Borel hierarchy ofUCF,UCF,the set of all continuous functions on the unit circle with uniformly convergent Fourier series. It turns out to be completeFσδ.F_{\sigma \delta }.Also we prove that anyGδσG_{\delta \sigma }set that includesUCFUCFmust contain a continuous function with divergent Fourier series.

We consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with a fully supported ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of Hölder continuous potentials. In particular, we apply our results to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains.

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)&lt;0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.