Continuous Solutions of the Functional Equation fn(x) = f(x)

1953 ◽  
Vol 5 ◽  
pp. 101-103 ◽  
Author(s):  
G. M. Ewing ◽  
W. R. Utz
Keyword(s):  
Functional Equation ◽  
Real Axis ◽  
Continuous Solutions ◽  
The Real ◽  
Real Functions ◽  
Image Position

In this note the authors find all continuous real functions defined on the real axis and such that for an integer n > 2, and for each x,

An Analogue of the Wave Equation and Certain Related Functional Equations

1969 ◽  
Vol 12 (6) ◽  
pp. 837-846 ◽  
Author(s):  
John A. Baker
Keyword(s):  
Functional Equation ◽  
Wave Equation ◽  
Functional Equations ◽  
Geometric Interpretation ◽  
Continuous Solutions ◽  
Difference Analogue ◽  
Image Position

Consider the functional equation(1)assumed valid for all real x, y and h. Notice that (1) can be written(2)a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

Graphs in the plane invariant under an area preserving linear map and general continuous solutions of certain quadratic functional equations

1985 ◽  
Vol 97 (2) ◽  
pp. 261-278 ◽  
Author(s):  
P. J. McCarthy ◽  
M. Crampin ◽  
W. Stephenson
Keyword(s):  
Functional Equation ◽  
Arbitrary Function ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Continuous Solutions ◽  
Linear Map ◽  
Area Preserving Maps ◽  
Image Position ◽  
Area Preserving

AbstractThe requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the formfor all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

scholarly journals A note on the finite Fourier transform

1959 ◽  
Vol 1 (1) ◽  
pp. 95-98
Author(s):  
James L. Griffith
Keyword(s):  
Fourier Transform ◽  
Real Axis ◽  
Exponential Type ◽  
Integral Function ◽  
Finite Fourier Transform ◽  
The Real ◽  
Almost Everywhere ◽  
Type C ◽  
Image Position

1. One of the best known theorems on the finite Fourier transform is:—The integral function F(z) is of the exponential type C and belongs to L2 on the real axis, if and only if, there exists an f(x) belonging to L2 (—C, C) such that ( Additionally, if f(x) vanishes almost everywhere in a neighbourhood of C and also in a neighbourhood of —C, then F(z) is of an exponential type lower than C.

An extension of the Ahlfors distortion theorem

1954 ◽  
Vol 50 (2) ◽  
pp. 261-265
Author(s):  
F. Huckemann
Keyword(s):  
Conformal Mapping ◽  
Real Axis ◽  
Upper Bounds ◽  
Distortion Theorem ◽  
Lower And Upper Bounds ◽  
The Real ◽  
Strip Domain ◽  
Parallel Strip ◽  
Image Position ◽  
Two Sides

1. The conformal mapping of a strip domain in the z-plane on to a parallel strip— parallel, say, to the real axis of the ζ ( = ξ + iμ)-plane—brings about a certain distortion. More precisely: consider a cross-cut on the line ℜz = c joining the two sides of the frontier of the strip domain (in these introductory remarks we suppose for simplicity that there is only one such cross-cut on that line), and denote by ξ1(c) and ξ2(c) the lower and upper bounds of ξ on the image in the ζ-plane. The theorem of Ahlfors (1), now classical, states thatprovided thatwhere a is the width of the parallel strip and θ(c) the length of the cross-cut.

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

scholarly journals On Koenigs' ratios for iterates of real functions

1969 ◽  
Vol 10 (1-2) ◽  
pp. 207-213 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Functional Equation ◽  
General Result ◽  
Real Function ◽  
Real Solutions ◽  
Real Functions ◽  
Image Position ◽  
Schröder Functional Equation

In a recent note, M. Kuczama [5] has obtained a general result concerning real solutions φ(x) on the interval 0 ≦ x < a ≦∞ of the Schröder functional equation providing the known real function satisfies the following (quite weak) conditions: f(x) is continuous and strictly increasing in ([0 a);(0) = 0 and 0 <f(x) <x for x ∈ (0, a); limx→0+ {f(x)/x} = s; and f(x)/x is monotonic in (0, a).

scholarly journals Roots, Schottky semigroups, and a proof of Bandt’s conjecture

2016 ◽  
Vol 37 (8) ◽  
pp. 2487-2555 ◽  
Author(s):  
DANNY CALEGARI ◽  
SARAH KOCH ◽  
ALDEN WALKER
Keyword(s):  
Real Axis ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Limit Set ◽  
The Real ◽  
Roots Of Polynomials ◽  
Image Position ◽  
Interior Points ◽  
Small Holes

In 1985, Barnsley and Harrington defined a ‘Mandelbrot Set’${\mathcal{M}}$for pairs of similarities: this is the set of complex numbers$z$with$0<|z|<1$for which the limit set of the semigroup generated by the similarities$$\begin{eqnarray}x\mapsto zx\quad \text{and}\quad x\mapsto z(x-1)+1\end{eqnarray}$$is connected. Equivalently,${\mathcal{M}}$is the closure of the set of roots of polynomials with coefficients in$\{-1,0,1\}$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ‘holes’ in${\mathcal{M}}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ‘exotic’ components of the space of (2-generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique oftrapsto construct and certify interior points of${\mathcal{M}}$, and use them to prove Bandt’s conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in${\mathcal{M}}$.

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART II

2012 ◽  
Vol 85 (2) ◽  
pp. 202-216 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f, g, h are real functions defined on an abelian group G. We solve this equation under the assumptions G=ℝ and f is continuous.

Representation Formulas for Integrable and Entire Functions of Exponential Type II

1991 ◽  
Vol 43 (1) ◽  
pp. 34-47 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Real Axis ◽  
Exponential Type ◽  
Entire Functions ◽  
Type Ii ◽  
The Real ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

We adopt the terminology and notations of [5]. If f ∈ Bτ is an entire function of exponential type τ bounded on the real axis then we have the complementary interpolation formulas [1, p. 142-143] andwhere t, γ are reals and

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