Characterization of the Fourier series of a distribution having a value at a point

Let be a periodic distribution of period . Let be its Fourier series. We show that the distributional point value exists and equals if and only if the partial sums converge to in the Cesàro sense as for each .

     

On the parallel evaluation of a sparse polynomial at a point

A new version of the Ruffini–Horner rule is presented for the evaluation of a polynomial of degree n at a point. In the PRAM model of parallel computation the new algorithm requires log n parallel steps with n/2+1 processors and the total number of arithmetic operations is n+⌈log₂(n+1)⌉ -1 multiplications and n additions. If the polynomial is sparse, i.e., the number of nonzero coefficients is k≪ n, then the total number of operations is at most k(⌈log n⌉- ⌊log k⌋)+2k+⌈log n⌉. Moreover, similarly to the customary Ruffini–Horner rule, the algorithm is backward numerically stable. In other words, the value provided by applying the algorithm in floating point arithmetic with machine precision μ coincides with the value taken on at the same point by a slightly perturbed polynomial. This revised version was published online in June 2006 with corrections to the Cover Date.     

On elastostatic boundary value problems with a conical boundary point

We study elastostatic boundary value problems with a conical boundary point by the method of integral equations. The equations of such problems are singular. In the case of a smooth surface, we construct a regularizer for these equations; in the case of a surface with a conical point, the regularizer is constructed in such a way as to ensure that the kernel of the regularized equation belongs to the class B and satisfies the assumptions of the Fredholm alternative theorem. We analyze the properties of elastic potentials in the case of a surface with a conical point.     

On the value of the critical point in fractal percolation

We derive a new lower bound p_c>0.8107 for the critical value of Mandelbrot's dyadic fractal percolation model. This is achieved by taking the random fractal set (to be denoted A_∞) and adding to it a countable number of straight line segments, chosen in a certain (nonrandom) way as to simplify greatly the connectivity structure. We denote the modified model thus obtained by C_∞, and write C_n for the set formed after n steps in its construction. Now it is possible, using an iterative technique, to compute the probability of percolating through C_n for any parameter value p and any finite n. For p=0.8107 and n=360 we obtain a value less than 10^?5; using some topological arguments it follows that 0.8107 is subcritical for C_∞ and hence (since _∞ dominates A_∞) for A_∞. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 332–345, 2001.     

On the value distribution of a certain type of differential polynomials

Some quantitative estimations on the value distribution of the functionafg ⁿ(n2) are obtained, whereg is a linear differential polynomial inf anda is a small function off.The research was partially supported by a U.G.C grant of Hong Kong     

On the value distribution of f a(f')~n

Let f be a transcendental meromorphic function,a a nonzero finite complex number,and n 2 a positive integer.Then f a(f')n assumes every complex value infinitely often.This answers a question of Ye for n = 2.A related normality criterion is also given.     

On the value distribution of the Matsumoto zeta-function

The Matsumoto zeta-function is defined by a polynomial Euler product and is a generalization of classical zeta-functions. In the paper a discrete limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Matsumoto zeta-function is proved.     

On analytic solutions of systems degenerate at a point

This paper deals with analytic solutions for a sufficiently wide class of systems of partial differential equations degenerate in one point.     

On the value distribution theory of elliptic functions

The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr ² (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , having coefficientsa(z) satisfyingT(r, a)=o(r ²) asr→∞, will the meromorphic functionh _Q (z)=Q(z, ?(z),...,?⁽ⁿ⁾(z)) either be identically zero or satisfyN(r, 1/h _Q )=o(r ²) asr→∞? In fact, we answer this question for rational functionsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z).     

On the cone of a finite dimensional compact convex set at a point

Four equivalent conditions for a convex cone in a Euclidean space to be an F_σ-set are given. Our result answers in the negative a recent open problem posed by Tam [5], characterizes the barrier cone of a convex set, and also provides an alternative proof for the known characterizations of the inner aperture of a convex set as given by Brønsted [2] and Larman [3].     

On the concept of point value in the infinite-dimensional realization theory

In this article, we study the effect of the chosen representation of a point value (and point evaluation) on the class of periodic signals realizable using a certain type of infinite-dimensional linear system. By suitably representing the point evaluation at the origin in a Hilbert space, we are able to give a complete characterization of its extensions. These extensions involve a new concept called δ-sequence, the use of which as an observation operator of an infinite-dimensional linear system is studied in this article. In particular, we consider their use in the realization of periodic signals. We also investigate how the use of δ-sequences affects the convergence properties of such realizations; we consider the rate and character of convergence and the removal of the Gibbs phenomenon. As still a further demonstration of the significance of the chosen concept of a point value, we discuss the use of distributional point values in the realization of periodic distributions. The possible applications of this work lie in regulator problems of infinite-dimensional control theory, as is indicated by the well-known internal model principle.