In this paper simple quartic Trigonometric polynomial blending functions, with a tension parameter, are presented. These type of functions are useful for constructing Trigonometric Bezier curves and surfaces, they can be applied to construct continuous shape preserving interpolation spline curves with shape parameters. To better visualize objects and graphics a tension parameter is included. In this work we constructed the Trigonometric Bezier curves followed by a construction of the shape preserving interpolation spline curves with local shape parameters and finally several numerical examples are presented such as open shape preserving interpolation curve, closed shape preserving interpolation curve and surfaces. As a direct application we computed the area surrounded by a closed curve.

By using the Trigonometric uniform splines of order 3 with a real tension factor, a numerical method is developed for solving a linear second order boundary value problems (2VBP) with Dirichlet, Neumann and Cauchy types boundary conditions. The moment at the knots is approximated by central finite-difference method. The order of convergence of the method and the theory is illustrated by solving test examples. Experimental results demonstrate that our method is more effective for the problems where the exact solution is Trigonometric or hyperbolic.

In this work, a new type of interpolation formula is introduced. These formulas can be an extension of the Lagrange interpolation formula. The error of this new type of interpolation is calculated. In order to display efficiency of the proposed formulas, three numerical examples are presented.

Let $S$ be a semigroup. In the present paper, we determine the complex-valued solutions $(f,g)$ of the functional equation$$g(x\sigma (y)) = g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\  x,y\in S,$$where $\sigma :S\rightarrow S$ is an automorphism that need not be involutive, and $\alpha \in \mathbb{C}$ is a fixed constant. Our results generalize and extend the ones by Stetk\ae r in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented.

Abram Haimovich Turetzkii [Uchenye Zapiski, 1 (149) (1959), 31-55 (translation in English in East J. Approx. 11 (2005), 337–359)] considered interpolatory quadrature rules which have the following form ¦2po ¦(x) w(x) dx» å2nv=o wv¦(xv), and which are exact for all Trigonometric polynomials of degree less than or equal to n. Maximal Trigonometric degree of exactness of such quadratures is 2n, and such kind of quadratures are known as quadratures of Gaussian type or Gaussian quadratures for Trigonometric polynomials. In this paper we prove some interesting properties of a special Gaussian quadrature with respect to the weight function wm(x) = 1+sin mx, where m is a positive integer.

The aim of this paper is to study the super stability problem of the mixed Trigonometric functional equations and the Hyers-Ulam-Rassias stability for a Jensen type functional equation.

In this note we discuss the convergence of greedy approximants for Trigonometric Fourier expansion in Lp(T), 1£p<2.