Cos Nx Research Articles

Parameter Estimation of Fractional Trigonometric Polynomial Regression Model

Trigonometric Polynomial Regression is a form of non-linear regression in which the relationship between the outcome variable and risk variable is Fractional modeled as 1/nth degree polynomial regression by combining the function of cos(nx) and sin(nx) on the value of natural numbers. The model was used to analyze the relationship between three continuous and periodic variables. Coefficients of the model were estimated using the Maximum Likelihood Estimate (MLE) method. From the results, the model obtained indicated that an increased in body mass index will increase the level of blood pressure while age may or may not have an influence on the blood pressure level. The values of the Coefficient of variation showed the variation in the dependent variable was well explained by the independent variables and the value of the adjusted coefficient of determination showed the model had a good fit with a high level of predictive power.

Simple expansions for sine and cosine

The student who has learned the binomial expansion and the formula for exp (ix), is shown how to obtain simple expansions for sin nx, cos nx, sinh nx, cosh nx, sin αx and cos αx for n any positive integer and α any real number.

On classes of everywhere divergent power series

We prove the everywhere divergence of series $$ \sum_{n=0}^\infty a_n e^{i\rho_n}e^{inx}, \quad\text{and}\quad \sum_{n=0}^\infty {(-1)}^{[\rho_n]}a_n \cos nx, $$ for sequences an and ρn satisfying some extremal conditions. These results generalize some well known examples of everywhere divergent power and trigonometric series.

On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Let f∈L2π be a real-valued even function with its Fourier series \(\frac {a_{0}}{2}+\sum\limits_{n=1}^{\infty}{a_{n}}\cos nx\), and let Sn(f,x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence {an} is decreasing and \(\lim\limits_{n\to\infty} a_{n} =0\), then \(\lim\limits_{n\to\infty} \|{f-S_{n}(f)}\|_{L}=0\) if and only if \(\lim\limits_{n\to\infty} a_{n}\log n=0\). Later, the monotonicity condition set on {an} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Korus further generalized the condition in the classical result to the so-called GM7 condition in real space. In this paper, we give a complete generalization to the complex space.

Evaluation of \\vint_{0}^{\\bf 2\\pi} \\exp (\\cos mx + \\cos nx)\\,\\hbox{d}x in Terms of Modified Bessel Functions of the First Kind For m, n \\in {\\open Z}, x \\in {\\open R}

The integral \\vint_{0}^{2\\pi} \\exp (\\cos mx + \\cos nx)\\,\\hbox{d}x is solved in terms of modified Bessel functions of the first kind for m, n \\in {\\open Z}, x \\in {\\open R}

Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums

It is proved that a trigonometric cosine series of the form ΣnEmphasis>=0/∞an cos(nx) with nonnegative coefficients can be constructed in such a way that all of its partial sums are positive on the real axis. It converges to zero almost everywhere and is not a Fourier-Lebesgue series. Some other properties of trigonometric series with nonnegative partial sums are also studied.

79.35 Summation and Properties of ∑ cos nx n k and ∑ sin nx n k

79.35 Summation and Properties of ∑ cos nx n k and ∑ sin nx n k

Level crossings of a random trigonometric polynomial with dependent coefficients

AbstractThis paper provides an asymptotic estimate for the expected number of K-level crossings of the random trigonometric polynomial g1 cos x + g2 cos 2x+ … + gn cos nx where gj (j = 1, 2, …, n) are dependent normally distributed random variables with mean zero and variance one. The two cases of ρjr, the correlation coeffiecient between the j-th and r-th coefficients, being either (i) constant, or (ii) ρ∣j−r∣ρ, j ≠ r, 0 < ρ < 1, are considered. It is shown that the previous result for ρjr = 0 still remains valid for both cases.

On the degree of aproximation of a periodic function f almost Riesz- means of its conjugate series

The present paper is concerned with the degree of approximation of certain functions belonging to the class Lip (p(t), p) by almost Riesz means. 1. Let f be a 27t-periodic function integrable Lp (p > 1) and let f >—’ + S (an Cos nx -f- bn Sin nx) (1.1) n=-l be its Fourier series.

Level crossings of a random trigonometric polynomial

This paper provides an asymptotic estimate for the expected number of K K -level crossings of the random trigonometric polynomial g 1 cos ⁡ x + g 2 cos ⁡ 2 x + … + g n cos ⁡ n x {g_1}\cos x + {g_2}\cos 2x + \ldots + {g_n}\cos nx , where g j ( j = 1 , 2 , … , n ) {g_j}(j = 1,2, \ldots ,n) are independent normally distributed random variables with mean μ \mu and variance one. It is shown that the result for K = μ = 0 K = \mu = 0 remains valid for any finite constant μ \mu and any K K such that ( K 2 / n ) → 0 ({K^2}/n) \to 0 as n → ∞ n \to \infty .

Limit cycles and chaos in equations of the pendulum type

It is proved that for sufficiently small ε the equation ẍ + sin x = ε/.x cos nx, n ϵ N where ε is a parameter, has exactly n − 1 coarse limit cycles (l.c.'s) in the region of oscillatory motions and no l.c.'s in the region of rotary motions (i.e., l.c.'s going round the phase cylinder). This result is used to study an equation of type (0.1) with time-periodic term on the right. The role of l.c.'s in the formation of quasi-attractors (q.a.'s) is demonstrated. A computer-generated description is given of the process by which g.a.'s with developed chaos are formed (for n = 3).

The integrability of generalized Garrett-Stanojević sums

In this paper we have defined the generalized Garrett-Stanojević cosine sums \[ h n ( x ) = ∑ p = 0 n S p r − 1 Δ r a p {h_n}\left ( x \right ) = \sum \limits _{p = 0}^n {S_p^{r - 1}{\Delta ^r}{a_p}} \] and have proved that under suitable conditions h n → h {h_n} \to h in the L 1 {L^1} -norm, where h ( x ) = a 0 / 2 + ∑ n = 1 ∞ a n cos ⁡ n x h\left ( x \right ) = {a_0}/2 + \sum \nolimits _{n = 1}^\infty {{a_n}\cos nx} . If r = 1 r = 1 , then h n ( x ) {h_n}\left ( x \right ) reduces to the modified cosine sums introduced by Rees and Stanojević.

An unsolvable Cousin problem

We construct a Laurent series ∑ n = − ∞ ∞ A n z n \sum \nolimits _{n = - \infty }^\infty {{A_n}{z^n}} , convergent in an annulus { ρ > | z | > 1 / ρ } \left \{ {\rho > \left | z \right | > 1/\rho } \right \} for some ρ > 1 \rho > 1 , which satisfies an algebraic differential equation (ADE) there, but such that neither of the series ∑ n = 0 ∞ A n z n \sum \nolimits _{n = 0}^\infty {{A_n}{z^n}} and ∑ n = − ∞ − 1 A n z n \sum \nolimits _{n = - \infty }^{ - 1} {{A_n}{z^n}} satisfies any ADE.

Some interesting coefficients

The aim of this article is to present some properties of the sequence This will be shown to generate a triangle, analogous to Pascal’s and which presents interesting properties. In particular, one application consists of deriving an easy method for expressing cos nx as a polynomial in cos x.

Convergence of certain cosine sums in a metric space — 𝐿

In this paper a generalization of a theorem of Garrett and Stanojevic [6] has been obtained.

Trigonometric series with monotone coefficients

Let {αn} be a monotonically decreasing sequence. Then each sequence {bn} such that bn ↓ 0, bn⩽αn, n=1, 2,..., is a sequence of Fourier-Lebesgue coefficients with respect to the system {cos nx} if and only if the sequence\(\sum\nolimits_{n = 1}^\infty {\frac{{{}^an}}{n}} \) converges.

A GEOMETRIC CONSTRUCTION OF THE DIRICHLET KERNEL*

AbstractIf sin (x/2) ≠ 0 then 1/2 + cos x + cos 2x + … + cos nx = sin (n + ½)x/2 sin (x/2). This relation is fundamental in the theory of Fourier series. We define a geometric construction in the euclidean plane whose analysis leads to this and related trigonometric identities.

Two Explicit Expressions for Cos nx

Pascal's triangle and matrix multiplication are used to express cos nx. Suitable for a trigonometry class familiar with elementary matrix work.

On the arithmetic mean of Fourier-Stieltjes coefficients

Let { a n } n = 0 ∞ \{ {a_n}\} _{n = 0}^\infty be the cosine Fourier-Stieltjes coefficients of the Borel measure μ \mu and { a 0 , ( a 1 + ⋯ + a n ) / n } n = 1 ∞ = { ( T a ) n } n = 0 ∞ \{ {a_0},({a_1} + \cdots + {a_n})/n\} _{n = 1}^\infty = \{ {(Ta)_n}\} _{n = 0}^\infty be the sequence of their arithmetic means. Then ∑ n = 0 ∞ ( T a ) n cos ⁡ n x \sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx} is a Fourier-Stieltjes series. Moreover, (a) ∑ n = 0 ∞ ( T a ) n cos ⁡ n x \sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx} is a Fourier series if and only if ( T a ) n → 0 {(Ta)_n} \to 0 at infinity or, equivalently, the measure μ \mu is continuous at the origin, (b) ∑ n = 1 ∞ ( T a ) n sin ⁡ n x \sum \nolimits _{n = 1}^\infty {{{(Ta)}_n}\sin nx} is a Fourier series if and only if the function x − 1 μ ( [ 0 , x ) ) {x^{ - 1}}\mu ([0,x)) is in L 1 [ 0 , π ] {L^1}[0,\pi ] . These results form the best possible analogue of a theorem of G. Goes, concerning arithmetic means of Fourier-Stieltjes sine coefficients, and improve considerably the theorems of L. Fejér and N. Wiener on the inversion and quadratic variation of Fourier-Stieltjes coefficients.

Commuting functions with no common fixed point

Introduction. Let f and g be continuous functions mapping the unit interval I into itself which commute under functional composition, that is, f(g(x))=g(f(V)) for all x in I. In 1954 Eldon Dyer asked whetherf and g must always have a common fixed point, meaning a point z in I for which f(z) =z=g(z). A. L. Shields posed the same question independently in 1955, as did Lester Dubins in 1956. The problem first appears in the literature in [15] as part of a more general question raised by J. R. Isbell. The purpose of this paper is to answer Dyer's question in the negative by the construction of a pair of commuting functions which have no fixed point in common. The connection between functions commuting and sharing fixed points appears in several areas of analysis. Perhaps the best-known example is the Markov-Kakutani theorem [11, p. 456], which states that a commuting family of continuous linear mappings of a conmpact convex subset of a linear topological space into itself has a common fixed point. The earliest relevant work on commuting functions was done in the 1920's by J. F. Ritt, who published several papers in which he investigated the algebraic properties of functional composition as a binary operation on the set of rational complex functions. His most important result from the modern standpoint was a characterizatioi-i of commuting (or permutable) rational functions [19]. He proved that iff and g are commuting polynomials, then, within certain homeomorphisms, either they are iterates of the same function (f= Fn and g = Fm for some F, n, in), both powers of x, or both must be Tchebycheff polynomials (defined by the relationship Tn(cos x)=cos nx). In either case a common fixed point may be shown to exist, so commuting polynomials have a common fixed point. The subject of commuting functions lay largely dormant until a 1951 paper by Block and Thielman [6] presented some new results on families of commuting polynomials and called attention to Ritt's earlier work. Their paper, together with the connection between commutativity and common fixed points found in other areas of mathematics, seems to have been the inspiration for the questions cited above. In the last few years a number of papers have been published on commuting