scholarly journals Delange's characterization of the sine function

1974 ◽  
Vol 15 (1) ◽  
pp. 66-68 ◽  
Author(s):  
Chin-Hung Ching ◽  
Charles K. Chui
Keyword(s):  
Entire Function ◽  
Analytic Continuation ◽  
Complex Plane ◽  
Real Line ◽  
Sine Function ◽  
The Real ◽  
Image Position

In [2], H. Delange gives the following characterization of the sine function.Theorem A. f(x)=sin x is the only infinitely differentiable real-valued function on the real line such that f'(O)= 1 andfor all real x and n = 0,1,2,….It is clear that, if f satisfies (1), then the analytic continuation of f is an entire function satisfyingfor all z in the complex plane. Hence f is of at most order one and type one. In this note, we prove the following theorem.

(LF)-spaces with absolute bases

1970 ◽  
Vol 67 (2) ◽  
pp. 283-286 ◽  
Author(s):  
G. Bennett ◽  
J. B. Cooper
Keyword(s):  
Complex Plane ◽  
Real Line ◽  
Convex Space ◽  
Inductive Limit ◽  
Locally Convex Space ◽  
The Real ◽  
Continuous Seminorm ◽  
Image Position ◽  
Locally Convex ◽  
Absolute Basis

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely aswhere ξk ∈K for each k. If this representation converges absolutely, i.e. iffor every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.

scholarly journals Yet another characterization of the sine function

1981 ◽  
Vol 4 (2) ◽  
pp. 371-381
Author(s):  
Robert Gervais ◽  
Lee A. Rubel
Keyword(s):  
Entire Function ◽  
Complex Plane ◽  
Exponential Type ◽  
Second Derivative ◽  
Sine Function ◽  
Number Of Zeros ◽  
Expository Paper

In this expository paper, it is shown that if an entire function of exponential type vanishes at least once in the complex plane and if it has exactly the same number of zeros (counting multiplicities) as its second derivative, then this function must take the formAsin(Bz+C).

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

On certain expansions of the solutions of the general Lamé equation

1942 ◽  
Vol 38 (4) ◽  
pp. 364-367 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Recurrence Relations ◽  
Characteristic Exponent ◽  
Periodic Functions ◽  
The Real ◽  
Lamé Equation ◽  
Arbitrary Complex ◽  
Principal Value ◽  
Image Position

1. In this paper I shall deal with the solutions of the Lamé equationwhen n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the formwhere θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.It is easy to obtain the system of recurrence relationsfor the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for whichk′ being the principal value of (1−k2)½

Lebesque measure zero subsets of the real line and an infinite game

1996 ◽  
Vol 61 (1) ◽  
pp. 246-249 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
Measure Zero ◽  
Cardinal Number ◽  
Minimal Cardinality ◽  
Combinatorial Characterization ◽  
The Real ◽  
Lebesque Measure ◽  
The Ideal

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:1. For each m, F(m) has at most m + 1 elements, and2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that1′. For each m, F(m) has at most Rm elements.We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

FIRST BETTI NUMBERS OF ABELIAN COVERINGS OF THE COMPLEX PROJECTIVE PLANE BRANCHED OVER LINE CONFIGURATIONS

2000 ◽  
Vol 09 (02) ◽  
pp. 271-284 ◽  
Author(s):  
IKUO TAYAMA
Keyword(s):  
Projective Plane ◽  
Real Line ◽  
General Position ◽  
Betti Numbers ◽  
Complex Projective Plane ◽  
The Real ◽  
Complex Projective

We prove the following results concerning the first Betti numbers of abelian coverings of CP2 branched over line configurations: (1) An estimate of the first Betti numbers. (2) A characterization of central and general position line configurations in terms of the first Betti numbers of abelian coverings. (3) The first Betti numbers of the abelian coverings of the real line configurations up to 7 components.

Sign in / Sign up

Export Citation Format

Share Document