Computational Methods and Function Theory最新文献

Abstract

Let (f = P[F]) denote the Poisson integral of F in the unit disc ({{mathbb {D}}}) with F absolutely continuous on the unit circle ({{mathbb {T}}}) and (dot{F}in L^p({{mathbb {T}}})) , where (dot{F}(e^{it}) = frac{d}{dt} F(e^{it})) . We show that for (pin (1,infty )) , the partial derivatives (f_z) and (overline{f_{bar{z}}}) belong to the holomorphic Hardy space (H^p({{mathbb {D}}})) . In addition, for (p=1) or (p=infty ) , (f_z) and (overline{f_{bar{z}}}in H^p({{mathbb {D}}})) if and only if (H(dot{F})in L^p({{mathbb {T}}})) , the Hilbert transform of (dot{F}) and in that case, we have (2izf_z=P[dot{F}+iH(dot{F})]) . Our main tools are integral representations of (f_z) and (f_{overline{z}}) in terms of (dot{F}) and M. Riesz’s theorem on conjugate functions. This simplifies and extends the recent results of Chen et al. [1] (J. Geom. Anal., 2021) and Zhu [17] (J. Geom. Anal., 2021).

In the present paper, we introduce and study the limit sets of the almost periodic functions f: ({{mathbb {R}}}rightarrow {{mathbb {C}}}). It is interesting, that (r=inf |f(x)|) and (R=sup |f(x)|) may be expressed in exact form. In particular, the formula for r coincides with the well known partition problem formula. We show that the ring (rle |z|le R) is the limit set of the almost periodic function f(x) (under some natural conditions on f). We obtain interesting applications to the partition problem in number theory and to trigonometric series theory. We extend classical results for periodic functions (estimation of Fourier coefficients and Bernstein’s theorem about absolute convergence of trigonometric series) to almost periodic functions. Using our results on almost periodic functions, we propose a new approach to interesting problems of the stability of motion. An application of our function theoretic results to the spectral problems in operator theory is studied as well. The results and corresponding computations are illustrated by several figures.

We study the best Möbius approximations (BMA) to convex and concave conformal mappings of the disk, including the special case of mappings onto convex polygons. The crucial factor is the location of the poles of the BMAs. Finer details are possible in the case of polygons through special properties of Blaschke products and the prevertices of the mapping function.

In this paper, the classification of rational functions whose Julia sets are Jordan arcs or curves, which started in (Carleson and Gamelin in Complex dynamics, Springer, Berlin, 1993; Steinmetz in Math Ann 307:531–541, 1997), will be completed. The method of proof is based on two quasi-conformal surgery procedures, which enables shifting the critical points in simply connected (super-)attracting and parabolic basins into a single critical point of highest possible multiplicity.

We give a combinatorial interpretation of vector continued fractions obtained by applying the Jacobi–Perron algorithm to a vector of (pge 1) resolvent functions of a banded Hessenberg operator of order (p+1). The interpretation consists in the identification of the coefficients in the power series expansion of the resolvent functions as weight polynomials associated with Lukasiewicz lattice paths in the upper half-plane. In the scalar case (p=1) this reduces to the relation established by P. Flajolet and G. Viennot between Jacobi–Stieltjes continued fractions, their power series expansion, and Motzkin paths. We consider three classes of lattice paths, namely the Lukasiewicz paths in the upper half-plane, their symmetric images in the lower half-plane, and a third class of unrestricted lattice paths which are allowed to cross the x-axis. We establish a relation between the three families of paths by means of a relation between the associated generating power series. We also discuss the subcollection of Lukasiewicz paths formed by the partial p-Dyck paths, whose weight polynomials are known in the literature as genetic sums or generalized Stieltjes–Rogers polynomials, and express certain moments of bi-diagonal Hessenberg operators.

In this paper, we give a complete characterization for meromorphic functions that share three distinct values (a,,b,,infty ) CM, with their difference operator (Delta _c f) or shift (f(z+c)). This provides a difference analogue of the corresponding results of Rubel-Yang, Mues-Steinmetz, and Gundersen. In particular, we prove that if an entire function f and its difference derivative (Delta _c f) share three distinct values (a,,b,,infty ) CM, then (fequiv Delta _c f). And our results show that the conjecture posed by Chen and Yi in 2013 holds for entire functions, and does not hold for meromorphic functions. Compared with many previous papers, our method circumvents the obstacle of the difference logarithmic derivative lemma for meromorphic functions of infinite order, since this method does not depend on the growth of the functions, but requires the knowledge of linear algebra and combinatorics.

For (alpha > 0), let (J_alpha ) be the class of all analytic functions f in the unit disk ({mathbb {D}}: = { z in {mathbb {C}}: |z| < 1 }) satisfying (f({mathbb {D}}) subset {mathbb {D}}) with the the angular derivative

For (a,zin mathbb {D}), let

Let (z_0 in {mathbb {D}}) be fixed. For (f in J_alpha ), we obtain the sharp estimate

with equality if and only if (f = sigma _{w_0}^{-1} circ sigma _{z_0}). Here (w_0 = (1-alpha k(z_0))/(alpha k(z_0) +1)). In case of (alpha k(z_0) > 1) we derive the estimate (|f'(z_0)| le k(z_0)/|1-z_0|^2). It is also sharp, however in contrast to the former case, there are no extremal functions in (J_alpha ). The lack of extremal functions is caused by the fact that (J_alpha ) is not closed in the topology of local uniform convergence in ({mathbb {D}}). Thus we consider the closure (bar{J}_alpha ) of (J_alpha ) and study (bar{V}_1(z_0, alpha ):= { f'(z_0): f in bar{J}_alpha }) which is the variability region of (f'(z_0)) when f ranges over (bar{J}_alpha ). We shall show that (partial bar{V}_1(z_0, alpha )) is a simple closed curve and (bar{V}_1(z_0, alpha )) is a convex and closed Jordan domain enclosed by (partial bar{V}_1(z_0, alpha )). Moreover, we shall give a parametric representation of (partial bar{V}_1(z_0, alpha )) and determine all extremal functions.