Search

This article deals with trigonometric functional equations on hypergroups. We describe the general continuous solution of sine and cosine addition formulas and a so-called sine-cosine functional equation on a locally compact hypergroup in terms of exponential functions, sine functions, and second-order generalized moment functions.

We consider a class of integral equations characterized by kernels of cosine and sine type and study their solvability. Moreover, we analyse the integral operator T, which is generating those equations, by identifying its characteristic polynomial, characterizing its invertibility, spectrum, Parseval type identity and involution properties. Additionally, a new convolution is here introduced, associated with T, for which we deduce a corresponding factorization property.

We construct the exact finite difference equation discretizations for the nonlinear differential equations whose solutions are the Jacobi cosine and sine functions. Our derivations clarify and extend previous work done on this topic.

In this paper, we introduce fractional cosine and sine transforms over finite fields. The basis for the definition of such transforms, which are respectively referred by the acronyms GFrCT and GFrST, is the recently proposed fractional Fourier transform over finite fields (GFrFT) [1] . More specifically, we use the eigenvectors of the finite field Fourier transform to construct eigenvectors of the finite field cosine and sine transforms. Such eigenvectors are then used in spectral expansions that allow to compute fractional powers of the finite field cosine and sine transform matrices.

In this paper we introduce some new modified cosine sums and then using these sums we study $L^{1}$-convergence of trigonometric cosine series.

In this paper, have study new results of half canonical cosine and sine transforms of generalized function. Keyword: Canonical transform, half canonical cosine transform, half canonical sine transform, testing function space.

In this paper the sine and cosine functional equations are considered and solved on arbitrary polynomial hypergroups in a single variable. The method of solution is based on spectral synthesis.

Hyperbolic sine and cosine functions on measure chains are introduced and their basic properties are established. A necessary and sufficient condition for (non)existence of focal points (generalized zeroes) of the hyperbolic cosine and of particular exponential functions on measure chains is found.

We introduce generalized convolutions of Stieltjes, Hilbert, and Fourier sine and cosine transforms and consider their applications to integral equations.

Some results concerning generalized Fourier sine- and cosine- transforms are discussed. The well known sine (or cosine) Fourier transform is an isometric mapping of L2 (0,∞) on itself (see [1]). It is interesting to consider the expression of a function with respect to sin(αξ + ϕ), where ϕ is a constant. Such an approach can be found in the papers of G.H. Hardy [2], R.G. Cooke [3], in [4] (the formula 7.10), and in a recent paper by A. Zilberglat and N. Lebedev [5]. In these works it was shown that an integrable function on (0, ∞) of a bounded variation over (0, ∞) can be repersented in the form of an integral of a hypergeometric function. In th Paper we consider generalized Fourier sine- and cosine- transforms of functions belonging to the space L2 (0, ∞). We shown the uniqueness and continuity of such an representation. Moreover, we obtain relations between formulae from the papers [2], [3] and [5].

Trigonometric matrix functions play a fundamental role in second order differential equation systems. This work presents an algorithm for computing the cosine matrix function based on Taylor series and the cosine double angle formula. It uses a forward absolute error analysis providing sharper bounds than existing methods. The proposed algorithm had lower cost than state-of-the-art algorithms based on Hermite matrix polynomial series and Padé approximants with higher accuracy in the majority of test matrices., This work has been supported by Universitat Politecnica de Valencia Grant PAID-06-011-2020.

Considering the characteristic of the angles, the sine and cosine values of which compose the discrete- sine and cosine wave, the new improved algorithm is presented. In the algorithm, the theory of setting Renminbi notes is used to selecting the basic angle sequence and the symmetry of trigonometric function is used to compute the sine and cosine values of the basic angle expressed as 2k?. Simulation results prove that the precision may reach 10^-6 under 5 iterations. Compared with other algorithms, The computational complexity of the algorithm is simple, the precise is higher. Therefore the algorithm is feasible for the NCO.

In this paper, we give connections between distribution cosine functions (defined in [10]) and almost-distribution cosine functions (introduced in [13]). We prove several equalities involving trigonometric convolution products and distribution cosine functions as well as some relations between distribution cosine functions and ultradistribution semigroups.

This chapter provides an overview of this book. The book presents the complete set of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) constituting the entire class of discrete sinusoidal unitary transforms, including their definitions, general mathematical properties, relations to the Karhunen-Loeve transform (KLT), with the emphasis on fast algorithms and integer approximations for their efficient implementations in the integer domain. The book covers various latest developments in DCTs and DSTs in a unified way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations, and integer approximations. For the DCT and DST to be viable, feasible, and practical, the fast algorithms are essential for their efficient implementation in terms of reduced memory, implementation complexity, and recursivity. Extensive definitions, principles, properties, signal flow graphs, derivations, proofs, and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines.

In this note, using the multisection series method, we establish the formulas for various power sums of sine or cosine functions.

By use of the closure operation for an odd integer set, the minimal number of real multiplications needed to compute length-2m four-type discrete cosine and sine transforms over the field Q, of rational numbers has been demonstrated.

In this note we give an elementary derivation for the power series expansions for the sine and the cosine function.

We define and study multivariate sine and cosine functions, symmetric with respect to the alternating group An, which is a subgroup of the permutation (symmetric) group Sn. These functions are eigenfunctions of the Laplace operator. They determine Fourier-type transforms. There exist three types of such transforms: expansions into corresponding sine-Fourier and cosine-Fourier series, integral sine-Fourier and cosine-Fourier transforms, and multivariate finite sine and cosine transforms. In all these transforms, alternating multivariate sine and cosine functions are used as a kernel.

(1987). The Derivatives of the Sine and Cosine Functions. The College Mathematics Journal: Vol. 18, No. 2, pp. 139-141.

(1968). Sine and Cosine Integrals. Optica Acta: International Journal of Optics: Vol. 15, No. 2, pp. 195-196.