Fast and slow points of Birkhoff sums

On the Polynomials Orthogonal on Regular Polygons

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

We consider continuous functions f which map the open unit disk D into the Riemann sphere W. For a point ζ on the unit circle C, we say that χ is a chord at ζ if χ is a chord of C having one endpoint at ζ and that Δ is a Stolz angle at ζ if Δ is a Stolz angle with vertex ζ. Suppose S denotes either a chord at ζ, a Stolz angle at ζ, or D.

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)<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.

On Hankel operators associated with a class of non-Toeplitz operators

Gibbs–Wilbraham oscillation related to an Hermite interpolation problem on the unit circle

We give a new algorithm for the construction of the unique superoptimal analytic approximant of a given continuous matrix-valued function on the unit circle, making use of exterior powers of operators in preference to spectral or {\em Wiener-Masani} factorizations.

On fully operator Lipschitz functions

As the winding number of unimodular function on R increases must the mass of the measure from which it arises (via the Fourier transform) increase?We work on the real numbers, R, ||a||m denotes the mass of measure p on R, and also we use \\p\\b to denote the same ||a||a where * is the standard Fourier transform.We will also use ||/||b in the case that / ^ p for any p M(T); in this case ||/||b denotes +00.Let / be real valued, continuous and nonlinear.H. Helson, J. P. Kahane [K, p. 87] ask if ||etr-^ ||b must exhibit at least a specific growth rate with respect to r. (The strong conjecture is ||etr-f ||b > K logr).Now in addition let / have a limit at +00.We ask a more specific question.Does there exist a function uj(r) with limit +00 at +00 and a constant K depending only on Sup / -Inf / such that ||eir^|| > Kw(r)?We motivate this question.Let g be a continuous unimodular function; write g = etJ* where / is continuous.If g has a finite winding number, then / has limits at 00, say L and M, and the winding number of g is (L -M)/2tt.Thus a yes answer to the qustion would show ||<7||b > w (winding number of g) where w(r) -* 00 as r - 00.THEOREM.For every 6 > 0 there exists a function uig such that w(r) -> +00 asr -* -|-oo that satisfies: If f is a real, nonconstant, continuous function on R that has a limit at -(-00.then (a) ||n.fc.||oo < 1-(b) p(mka) > 1. (c) \p(prnksm~ki)\ < e~Q forp Pfc_x and s < t.Then \\p\\M > y/Q/A.

We study the operator algebra associated with a self-mapping ϕ on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l2(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).