双解析函数的一般复合边值问题关于边界曲线的稳定性

开口弧段Γ上的双解析函数的Riemann边值问题与单位圆周L上双解析函数的Hilbert边值问题复合而成的一般复合边值问题,当L与Γ发生微小的光滑摄动后,借助于推广的拉甫伦捷夫近似于圆的共形映射,将星形域映为单位圆域,从而得出摄动后的问题的解的表达式,同时讨论了解的稳定性情况,并给出误差估计.     

On solving the second basic three-element boundary-value Carleman-type problem in classes of bianalytic functions in a disk

In this paper, we investigate one of the basic Carleman-type boundary-value problems for bianalytic functions. We obtain a constructive method for solving the problem in the case of circular domain. We establish that solving the problem can be reduced to a sequential solving two generalized Carleman-type problems for analytic functions in a disk. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 413–426, July–September, 2006.     

A probabilistic property of Katsuura's continuous nowhere differentiable function

Katsuura's example of a continuous nowhere differentiable function is based on a fixed-point method. In this paper we use probability theory to study the increments of Katsuura's function.     

An example of bad convex function

An example of a convex function having the gradient at each point (x, 0, ..., 0),x>0, which does not converge, whenx tends to zero, is given.The author would like to thank Professor Giannessi for his help, which led to an improved version of this paper.     

Mixed boundary value problem for some pairs of metaanalytic function and analytic function

In this article, some mixed boundary value problems (BVPs) on the unit circumference for some pairs of a metaanalytic function and an analytic function are discussed. Using the relationship between metaanalytic function and polyanalytic function, the expression of solution and the condition of solvability for the problem are obtained by reducing the problem to an equivalent system of a Haseman BVP for analytic function and a Hilbert BVP for analytic function. Copyright © 2008 John Wiley & Sons, Ltd.     

The divisor of a generalized analytic function

For generalized analytic functions, we define the order of a zero and prove the following division theorem: if the orders of zeros of the numerator majorize those of the denominator, then the fraction is a generalized analytic function. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 655–661, May, 1997. Translated by V. E. Nazaikinskii     

An equivalence for weighted integrals of an analytic function and ist derivative

In this paper we introduce the class of differentiable weights ω in the unit disc ?? such that where L is a positive constant. The main result in this paper asserts that if ω is one of these weights, then the equivalence holds for all 0 < p < ∞, 0 < q ≤ ∞and f an analytic function in ??. Our results improve others due to Aleman, Siskakis, and Stevi?. We also prove two results on harmonic conjugate functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuation of separately analytic functions defined on part of a domain boundary

Suppose that D ? ?ⁿ is a domain with smooth boundary ?D, E ? ?D is a boundary subset of positive Lebesgue measure mes(E) > 0, and F ? G is a nonpluripolar compact set in a strongly pseudoconvex domain G ? ?^m. We prove that, under some additional conditions, each function separately analytic on the set X = (D×F)∪(E× G) can be holomorphically continued into the domain where ω* is the P-measure and ω _in ^* is the inner P-measure.     

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

We discuss Totik’s extension of the classical Bernstein theorem on polynomial approximation of piecewise analytic functions on a closed interval. The error of the best uniform approximation of such functions on a compact subset of the real line is studied.     

Continuation of separately analytic functions defined on part of the domain boundary

Let D ? ? ⁿ be a domain with smooth boundary ?D, let E??D be a subset of positive Lebesgue measure mes(E) > 0, and let F ? G be a nonpluripolar compact set in a strongly pseudoconvex domain D ? ? ^m . We prove that, under an additional condition, each function separately analytic on the set X = (D × F) ∪ (E × G) has a holomorphic contination to the domain $\rlap{--} X = \{ (z,w) \in D \times G:\omega _{in}^ * (z,E,D) + \omega ^ * (w,F,D) < 1\} $ , where ω* is the P-measure and ω*_in is the interior P-measure.     

复合解析函数族分担亚纯函数的正规性

本文推广了Bergweiler的一个正规定则:设α(z)和F分别是区域D上的非常数解析函数与解析函数族,R(z)是一个次数不低于2的有理函数.如果对族F中函数f(z)和g(z),Rof(z)和Rog(z)分担α(z)IM,并且下述条件之一成立:(1)对任意z0∈D,R(z)-α(z0)有至少两个不同的零点或极点;(2)存在z0∈D使得R(z)-α(z0):=P(z)Q(z)仅有一个零点(或极点)β0,同时k=lp(或k=lq),其中l和k分别是f(z)-β0和α(z)-α(z0)在z0处的零点重数,P(z)和Q(z)分别是次数为p和q的互质的多项式,并且α(z0)∈C∪{∞}.那么F在D内正规.     

Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

In this article, we reconsider the mixed boundary value problem on the unit circle for a pair of metaanalytic and analytic functions as in Du and Wang (2008) [9]. By adopting appropriate transformations, we convert the problem into two independent boundary value problems for analytic functions. We then obtain expressions of solution and condition of solvability for the mixed boundary value problem. The forms of the solutions and the condition of solvability here are rather dissimilar to those in Du and Wang (2008) [9]. But the equivalence is established at the end of this article.     

On a new class of analytic function derived by a fractional differential operator

Making use of the fractional differential operator, we impose and study a new class of analytic functions in the unit disk (type fractional differential equation). The main object of this paper is to investigate inclusion relations, coefficient bound for this class. Moreover, we discuss some geometric properties of the fractional differential operator.     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

An existence criterion for a smooth function under constraints

In this paper, we study conditions for the existence of a function satisfying constraints on its value at each point and on its derivative.     

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function (x;α,h)≡exp(-[α²/h²]x²) where α is a user-choosable constant. Unfortunately, computing the coefficients of (x-jh;α,h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C_j(x;α,h) is defined by the set of linear combinations of the Gaussians such that C_j(kh)=1 when k=j and C_j(kh)=0 for all integers . We show that the cardinal functions for the uniform grid are C_j(x;h,α)=C(x/h-j;α) where C(X;α)≈(α²/π)sin(πX)/sinh(α²X). The relative error is only about 4exp(-2π²/α²) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h→0, but only to an “error saturation” proportional to exp(-π²/α²). Because the error in our approximation to the master cardinal function C(X;α) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f(x). The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.     

An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions

The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.     

Entire functions, analytic continuation, and the fractional parts of a linear function

The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R₀, the following inequality holds: