A Functional Equation for the Cosine

1968 ◽  
Vol 11 (3) ◽  
pp. 495-498 ◽  
Author(s):  
PL Kannappan
Keyword(s):  
Functional Equation ◽  
Measurable Function ◽  
Trivial Solution ◽  
Real Variable ◽  
Real Constant ◽  
Real Numbers ◽  
Measurable Solution ◽  
Image Position ◽  
Complex Valued ◽  
Cosine Functions

It is known [3], [5] that, the complex-valued solutions of(B)(apart from the trivial solution f(x)≡0) are of the form(C)(D)In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],(E)In this paper, the functional equation(P)where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.

The definition of measure in function space

1950 ◽  
Vol 46 (1) ◽  
pp. 19-27
Author(s):  
H. R. Pitt
Keyword(s):  
Function Space ◽  
Real Variable ◽  
Fundamental Result ◽  
Real Numbers ◽  
Real Functions ◽  
Definition Of ◽  
Image Position ◽  
The Right ◽  
Positive Integers

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).

Positive-moment problems in abstract measure spaces

1975 ◽  
Vol 78 (3) ◽  
pp. 461-469
Author(s):  
H. P. Rogosinski
Keyword(s):  
Moment Problem ◽  
Measurable Function ◽  
Measure Space ◽  
Moment Problems ◽  
Real Numbers ◽  
Index Set ◽  
Almost Everywhere ◽  
Measure Spaces ◽  
Abstract Measure ◽  
Image Position

In this paper we continue the investigation of positive-moment problems, begun in (4). For an arbitrary index set A we consider a family (fα)α ∈ A of measurable real-valued functions on a measure-space (X, µ). We suppose throughout thatwhere (Xm) is an increasing sequence of measurable subsets of X and where, for each α in A and each m, fα is µ-integrable over Xm. Let (сα)α ∈ A be a given family of real numbers. We consider the following restricted positive-moment problem: does there exist a measurable function g on X such that 0 ≤° g ≤° 1 and such thatfor every α in A? (Here the symbol ‘≤°’ indicates that the relation ≤ holds almost everywhere with respect to µ on X. Symbols ‘ = °, <°, …’ are used similarly.) If such a g exists we call (сα)α ∈ A a moment family for the problem:

Generalized Variation and Functions of Slow Growth

1988 ◽  
Vol 40 (1) ◽  
pp. 55-85 ◽  
Author(s):  
Robert D. Berman
Keyword(s):  
Bounded Variation ◽  
Poisson Kernel ◽  
Slow Growth ◽  
Real Variable ◽  
Cauchy Kernel ◽  
Formal Integration ◽  
Carry Over ◽  
Image Position ◽  
Generalized Variation ◽  
Complex Valued

Many of the basic results of HP theory on the disk Δ = {|z| < 1} are proved using the Cauchy-Stieltjes representation1.1and the Poisson-Stieltjes representation1.2Here, μ:R → C is a complex-valued function of a real variable that is of bounded variation on [0, 2π] such that μ(t + 2π) = μ(t) + μ(2t) — μ(0), t ∊ R,is the Cauchy kernel, andis the Poisson kernel. It is therefore natural to generalize these representations in such a way that some of the basic properties and results carry over. Such a generalization occurs when the assumption that μ is of bounded variation on [0, 2μ] is replaced by the requirement that it is measurable and bounded on [0, 2μ] (cf. [9]). The integrals in (1.1) and (1.2) are then defined by a formal integration by parts. After some preliminaries in Section 2, we catalogue a variety of results which remain valid in Section 3.

HEAT KERNEL METHOD FOR THE LEVI-CIVITÁ EQUATION IN DISTRIBUTIONS AND HYPERFUNCTIONS

2015 ◽  
Vol 92 (1) ◽  
pp. 77-93
Author(s):  
JAEYOUNG CHUNG ◽  
PRASANNA K. SAHOO
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Kernel Method ◽  
Functional Inequality ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Real Numbers ◽  
Positive Real ◽  
Image Position ◽  
Ulam Stability Problem

Let$G$be a commutative group and$\mathbb{C}$the field of complex numbers,$\mathbb{R}^{+}$the set of positive real numbers and$f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$. In this paper, we first consider the Levi-Civitá functional inequality$$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$where${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$is a symmetric decreasing function in the sense that${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$for all$0<t_{1}\leq t_{2}$and$0<s_{1}\leq s_{2}$. As an application, we solve the Hyers–Ulam stability problem of the Levi-Civitá functional equation$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$in the space of Gelfand hyperfunctions, where$u,v,w,k$are Gelfand hyperfunctions,$S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$, and$\circ$,$\otimes$,${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$and${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$denote pullback, tensor product and the spaces of bounded distributions and bounded hyperfunctions, respectively.

A convolution-integral representation for a class of linear operators

1972 ◽  
Vol 71 (1) ◽  
pp. 51-60 ◽  
Author(s):  
K. Rowlands
Keyword(s):  
Shift Operator ◽  
Conjugate Function ◽  
Real Variable ◽  
Linear Operators ◽  
Complex Conjugate ◽  
Convolution Integral ◽  
Complex Vector ◽  
Shift Operators ◽  
Image Position ◽  
Complex Valued

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formulaA linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:(a) for each x in , the complex-conjugate function x* is in ;(b) both and \ are invariant under the shift operators;(c) every shift operator commutes with T.(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)

A HYPERSTABILITY RESULT FOR THE CAUCHY EQUATION

2013 ◽  
Vol 89 (1) ◽  
pp. 33-40 ◽  
Author(s):  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Normed Space ◽  
Functional Equations ◽  
Cauchy Equation ◽  
Cauchy Functional Equation ◽  
Real Numbers ◽  
Formula One ◽  
Image Position

AbstractWe prove a hyperstability result for the Cauchy functional equation$f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function$f$, mapping a normed space${E}_{1} $into a normed space${E}_{2} $, and for all real numbers$r, s$with$r+ s\gt 0$one of the following two conditions must be valid:$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem.

On a Relation between the “Square” Functional Equation And The “Square” Mean-Value Property

1971 ◽  
Vol 14 (2) ◽  
pp. 161-165 ◽  
Author(s):  
Hiroshi Haruki
Keyword(s):  
Functional Equation ◽  
Real Variable ◽  
Mean Value ◽  
Geometric Meaning ◽  
Mean Value Property ◽  
Image Position

We consider the following functional equation1where ƒ = ƒ(x, y) is a real-valued function of two real variables x, y on the whole xy-plane and t is a real variable.With regard to the geometric meaning of (1), the equation is called the “square” functional equation.

Functional Equations, Distributions and Approximate Identities

1990 ◽  
Vol 42 (4) ◽  
pp. 696-708 ◽  
Author(s):  
John A. Baker
Keyword(s):  
Functional Equation ◽  
Lebesgue Measure ◽  
Functional Equations ◽  
Schwartz Distributions ◽  
Integrable Functions ◽  
Approximate Identities ◽  
The Subject ◽  
Image Position ◽  
Complex Valued

The subject of this paper is the use of the theory of Schwartz distributions and approximate identities in studying the functional equationThe aj’s and b are complex-valued functions defined on a neighbourhood, U, of 0 in Rm, hj. U → Rn with hj(0) = 0 and fj, g: Rn → C for 1 ≦ j ≦ N. In most of what follows the aj's and hj's are assumed smooth and may be thought of as given. The fj‘s, b and g may be thought of as the unknowns. Typically we are concerned with locally integrable functions f1, … , fN such that, for each s in U, (1) holds for a.e. (almost every) x ∈ Rn, in the sense of Lebesgue measure.

Cosine Representations of Abelian *-Semigroups and Generalized Cosine Operator Functions

1978 ◽  
Vol 30 (03) ◽  
pp. 474-482 ◽  
Author(s):  
G. D. Faulkner ◽  
R. W. Shonkwiler
Keyword(s):  
Functional Equation ◽  
Linear Operators ◽  
Bounded Operators ◽  
Real Numbers ◽  
Bounded Linear ◽  
Cosine Operator ◽  
D’Alembert’S Functional Equation ◽  
Operator Functions ◽  
Cosine Operator Functions ◽  
Image Position

In the following R will denote the real numbers, for a Hilbert space H, B(H) and L(H) will denote the collections of bounded linear operators on H and linear, but not necessarily bounded, operators on H respectively. Cosine Operator Functions, namely functions C:R ⟶ B(H) which satisfy D'Alembert's functional equation (1) and (2)

Möbius transformations in stability theory

1970 ◽  
Vol 68 (1) ◽  
pp. 143-151 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Characteristic Equation ◽  
Stability Theory ◽  
Trivial Solution ◽  
Real Variable ◽  
Continuous Functions ◽  
Asymptotically Stable ◽  
Image Position ◽  
Special Case

1. Introduction: Consider the system of ordinary differential equationswhere the unknown x(t) is a complex m-vector, t is a real variable, D is the operator d/dt and a0, …, an are complex m × m matrices whose elements are continuous functions of t, x, Dx, …, Dn−1x. Furthermore, det a0 ╪ 0. In the special case when a0, …, an are constant matrices the trivial solution x = 0 is asymptotically stable if and only if all the roots of the characteristic equation det f(ζ) = 0 have negative real parts, where

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