A Continuous Periodic Function has every Chord Twice

1967 ◽  
Vol 74 (7) ◽  
pp. 833
Author(s):  
J. B. Diaz ◽  
F. T. Metcalf
Keyword(s):  
Periodic Function ◽  
Continuous Periodic Function

On the existence of periodic solutions of a certain third-order differential equation

1960 ◽  
Vol 56 (4) ◽  
pp. 381-389 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Initial Conditions ◽  
Continuous Periodic Function ◽  
Third Order ◽  
Third Order Differential Equation ◽  
Existence Of Periodic Solutions ◽  
Image Position

In this paper we shall be concerned with the differential equationin which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:under which every solution x(t) of (1) satisfieswhere t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),or, in the case (3),In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).

scholarly journals Absolutely convergent Fourier series and generalized Zygmund classes of functions

2009 ◽  
Vol 104 (1) ◽  
pp. 124
Author(s):  
Ferenc Móricz
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Fourier Coefficients ◽  
Sufficient Conditions ◽  
Modulus Of Smoothness ◽  
Slowly Varying Function ◽  
Continuous Periodic Function ◽  
Convergent Fourier Series ◽  
Period 2 ◽  
Order Of Magnitude

We investigate the order of magnitude of the modulus of smoothness of a function $f$ with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that $f$ belongs to one of the generalized Zygmund classes $(\mathrm{Zyg}(\alpha, L)$ and $(\mathrm{Zyg} (\alpha, 1/L)$, where $0\le \alpha\le 2$ and $L= L(x)$ is a positive, nondecreasing, slowly varying function and such that $L(x) \to \infty$ as $x\to \infty$. A continuous periodic function $f$ with period $2\pi$ is said to belong to the class $(\mathrm{Zyg} (\alpha, L)$ if 26740 |f(x+h) - 2f(x) + f(x-h)| \le C h^\alpha L\left(\frac{1}{h}\right)\qquad \text{for all $x\in \mathsf T$ and $h>0$}, 26740 where the constant $C$ does not depend on $x$ and $h$; and the class $(\mathrm{Zyg} (\alpha, 1/L)$ is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of $f$ are all nonnegative.

scholarly journals Affine Bessel sequences and Nikishin’s example

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 963-966 ◽  
Author(s):  
V.A. Mironov ◽  
A.M. Sarsenbi ◽  
P.A. Terekhin
Keyword(s):  
Hilbert Space ◽  
Periodic Function ◽  
Spectral Theory ◽  
Continuous Periodic Function ◽  
Affine Systems ◽  
Convergence In Measure ◽  
Affine System ◽  
Bessel Sequences

We study affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. We construct a non-Besselian affine system fun(x)g1 n=0 generated by continuous periodic function u(x). The result is based on Nikishin?s example concerning convergence in measure. We also show that affine systems fun(x)g1 n=0 generated by any Lipchitz function u(x) are Besselian.

scholarly journals Analytical solutions of a particular Hill's differential system

2019 ◽  
Vol 11 (1) ◽  
pp. 121-129
Author(s):  
Nicolae MARCOV
Keyword(s):  
Periodic Function ◽  
Numerical Solution ◽  
Explicit Form ◽  
Analytical Solutions ◽  
Fundamental Matrix ◽  
Continuous Periodic Function ◽  
Periodic Term ◽  
First Order ◽  
Analytical Functions ◽  
Periodic Equation

Consider a second order differential linear periodic equation. This equation is recast as a first-order homogeneous Hill’s system. For this system we obtain analytical solutions in explicit form. The first solution is a periodic function. The second solution is a sum of two functions; the first is a continuous periodic function, but the second is an oscillating function with monotone linear increasing amplitude. We give a formula to directly compute the slope of this increase, without knowing the second numerical solution. The periodic term of second solution may be computed directly. The coefficients of fundamental matrix of the system are analytical functions.

The limit-point, limit-circle nature of rapidly oscillating potentials

1977 ◽  
Vol 76 (3) ◽  
pp. 183-196 ◽  
Author(s):  
F. V. Atkinson ◽  
M. S. P. Eastham ◽  
J. B. McLeod
Keyword(s):  
Periodic Function ◽  
Limit Point ◽  
Continuous Periodic Function ◽  
Limit Circle

SynopsisThe Weyl limit-point, limit-circle nature of the equation y″(x)–q(x)y(x) = 0(0≦ x< ∞) is analysed When q(x) has the form q(x) = xαp(xβ), where α and β are positive constants and p(t) is a continuous periodic function of t.

scholarly journals Primeable entire functions

1973 ◽  
Vol 51 ◽  
pp. 123-130 ◽  
Author(s):  
Fred Gross ◽  
Chung-Chun Yang ◽  
Charles Osgood
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
Rational Function ◽  
Entire Functions ◽  
Left And Right

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

Projective actions, invariant sigma-curves and quadratic functional equations

1985 ◽  
Vol 98 (2) ◽  
pp. 195-212 ◽  
Author(s):  
Patrick J. McCarthy
Keyword(s):  
Functional Equation ◽  
Periodic Function ◽  
Functional Equations ◽  
Continuous Solution ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Continuous Solutions ◽  
Linear Map

AbstractThe quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.

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