Consider a collection $λ_1<...<λ_N$ of distinct positive integers and the quantities $$ M_1 = M_1(λ_1,...,λ_N) =
\max_{0\le x \le 2π} |\sum_{j=1}^N \sin{λ_j x}| $$ and $$ M_2 = M_2(λ_1,...,λ_N) = - \min_{0\le x \le 2π}
\sum_{j=1} \cos{λ_j x}. $$ Prompted by a discussion with G. Benke we prove that collections of frequencies $λ_j$ which have $M_1 = o(N)$ or $M_2 = o(N)$ are unstable, in the sense that one can perturb the $λ_j$ by one each and get $M_1 \ge c N$ and $M_2 \ge c N$.
In this paper we define canonical sine and cosine transform, convolution operations, prove convolution theorems in space of integrable functions on real space. Further, obtain some results require to construct the spaces of integrable Boehmians then extend this canonical sine and canonical cosine transforms to space of integrable Boehmians and obtain their properties.
In this work, series expansions in terms of Bessel functions of the first kind are given for the sine and cosine integrals. These representations differ from many of the known Neumann-type series expansions for the sine and cosine integrals, which contain non-integer order or quadratic Bessel function terms. In addition, using the theory of Euler sums we are able to obtain some closed form evaluations of integrals involving Bessel functions and the sine and cosine integrals.
In this paper, we provide a concise proof of Oppenheim's double inequality relating to the cosine and sine functions. In passing, we survey this topic.
In this paper we establish the stability of the functional equation $$f(x-y)=f(x)g(y)+g(x)f(y)+h(x)h(y)),\;\; x,y \in G, $$where $G$ is an abelian group.
The non-elementary integrals $\mbox{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α>β+1$ and $\mbox{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α>2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral $\mbox{Ei}_{β,α}=\int (e^{λx^β}/x^α) dx, β\ge1, α>β+1$ is expressed in terms of $_{2}F_2$. The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term.
A description of eigensubspaces of the cosine and sine operators is presented. The spectrum of each of these two operator consists of two eigenvalues (1,\,-1) and their eigensubspaces are infinite--dimensional. There are many possible bases for these subspaces, but most popular are bases constructed from the Hermite functions. We present other "bases" which are not discrete orthogonal sequences of vectors, but continuous orthogonal chains of vectors. Our work can be considered a continuation and further development of results in \textit{Self-reciprocal functions} by Hardy and Titchmarsh: Quart…
Translated from the Latin original, "Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur" (1777). E655 in the Enestrom index. Euler looks at the binomial expansion $(1+x)^n$ for $x=\cos ϕ+i\sinϕ$. He does $n=$ positive and negative integers, and $n={1/2}, -{1/2}$, and particular values of $ϕ$.
We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing certain relations between the Fourier transforms and their Hilbert transforms.
We present a new family of sine polynomials that are nonnegative for all $x$ in $[0,π]$. We also characterize all nonnegative sine polynomials of degree 3 and all nonnegative cosine polynomials of degree 2. In the latest version, typos in (1.4) and (1.6) are corrected (with >= replaced by <=).
In this paper we prove asymptotic formulas for the $L^p$ norms of $P_n(θ)=\prod_{k=1}^n (1-e^{ikθ})$ and $Q_n(θ)=\prod_{k=1}^n (1+e^{ikθ})$. These products can be expressed using $\prod_{k=1}^n \sin\Big(\frac{kθ}{2}\Big)$ and $\prod_{k=1}^n \cos\Big(\frac{kθ}{2}\Big)$ respectively. We prove an estimate for $P_n$ at a point near where its maximum occurs. Finally, we give an asymptotic formula for the maximum of the Fourier coefficients of $Q_n$.
The non-elementary integrals $\text{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α\leβ+1$ and $\text{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α\le2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric functions $_{1}F_2$ and $_{2}F_3$, and their asymptotic expressions for $|x|\gg1$ are also derived. The integrals of the form $\int [\sin^n{(λx^β)}/(λx^α)] dx$ and $\int [\cos^n{(λx^β)}/(λx^α)] dx$, where $n$ is a positive integer, are expressed in terms $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$, and then evaluated. $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$ are also…
Generalised definitions of exponential, trigonometric sine and cosine and hyperbolic sine and cosine functions are given. In the lowest order, these functions correspond to ordinary exponential, trigonometric sine etc. Some of the properties of the generalised functions are discussed. Importance of these functions and their possible applications are also considered.
In the paper, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor's series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents an explicit formula of specific partial Bell polynomials at a sequence of derivatives of a function contain…
Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family, allows one to introduce symmetrized and antisymmetrized mult…
In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in establishing some recent results on nonnegative trigonometric polynomials. See, for example, [2], [4] and [5]. It continues to play an essential role in several ongoing projects. The algorithm, however, cannot handle general trigonometric polynomials that involve both cosine and sine terms. Some ad hoc methods to deal with such polynomials have been suggested in [3], …
The purpose of this paper is to define generalized Apostol--Bernoulli polynomials with including a new cosine and sine parametric type of generating function using the quasi-monomiality properties and trigonometric functions. In this study, the Apostol-Bernoulli polynomials with three variable are defined with two new generating functions cosine and sine parameters. Then, we investigate multiplicative and derivative operators, diffrential equations, some summation formulas and partial differential equations for these polynomials. Moreover, we introduce Gould--Hopper--Apostol--Bernoulli type po…
In this work, a Vietoris type theorem for the positivity of sine and cosine sum for a particular sequence of real numbers is provided. In this connection, the positivity of a particular type of sine sum involving ratio of some parameters is given, which is new in the literature. Various new results that follow from the Vietoris type theorem include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums for orthogonal polynomials. An open problem is also provided for the partial sums of the generalized polylogarithm.
