Received: 21 September 2019 Revised: 13 November 2019 Accepted: 14 November 2019 DOI: 10.1002/zamm.201900262 ORIGINAL PAPER On algebraic trigonometric integro splines Salah Eddargani 1 Abdellah Lamnii 1 Mohamed Lamnii 2 1 University Hassan First, FST, MISI Laboratory, Settat, Morocco 2 University Mohammed I, FSO, LANO Laboratory, Oujda, Morocco Correspondence Abdellah Lamnii, University Hassan First, FST, MISI Laboratory, Settat, Morocco. Email: alamnii@gmail.com In this paper, we present a new kind of quadratic approximation operator reproducing of both algebraic and trigonometric functions. It is called integro quadratic splines interpolant, which agree with the given integral values of a univariate real-valued function over the same intervals, rather than the functional values at the knots. Effi- cient approximations of fractional integrals and fractional Caputo derivatives based on this interpolant, are constructed and well studied. The general approximation error is studied too, and the super convergence property is also derived when the interval is equally partitioned. Numerical examples illustrate that our method is very effective and our quadratic algebraic trigonometric integro spline has higher approximation ability than others. KEYWORDS algebraic trigonometric splines, error bound, fractional caputo derivatives, fractional integrals, integro spline quasi-interpolant 1 INTRODUCTION In numerical analysis, interpolation is a mathematical operation that consist of replacing a curve or a function  with a simpler curve (or function) , but which coincides with the first one at a finite number of points (or values) given at the beginning, i.e.  (  )= (  ),  =0, … , . Recently, spline functions, which are well known as piecewise polynomials pieced together at the knots by certain smoothness conditions, have attracted the interest of researchers due to their properties and usefulness in mathematical approximation theory. Thus, they are often applied on this topic (for more details see [1–4] and references therein). In some cases, we deal with phenomena which only involve the integral value of a function  . The question is, if we know the integral values of the function, then how we can use these values to construct quadratic splines. This kind of approximation was arising in many works in literature. In [5] and [6], Behforooz developed two kinds of integro splines quasi-interpolant, cubic and quintic operators respectively. The two operators introduced by Behforooz need various end conditions and solving a tridiagonal system of linear equations. Then, to avoid solving linear system of equation, the authors in [7], developed cubic integro splines quasi-interpolant without solving any system of equations. Lang and Xu in [8], constructed an integro quartic spline and its approximation properties. They showed that the integro quartic spline possesses superconvergence orders in approximating function values and second order derivatives values at the knots, but a phenomenon of superconvergence remains without answer. To this end, T. Zhanlav and R. Mijiddorj in [9], also constructed a local integro quartic splines quasi-interpolants. Some others authors have devoted much intention to construct this kind of operators, for more details, can see [10–15]. In the present work, we discuss the integro interpolation problem by using algebraic trigonometric quadratic B-splines. The operator presented here, has some advantages compared to the other existing methods. More precisely, our operator has the abilities of reproducing algebraic trigonometric function (i.e.  {1, sin(), cos()}). Thus, and as we will mention later, this operator has a super-convergence phenomena when the step size is uniform. Compared to other existing approaches, the present Z Angew Math Mech. 2019;e201900262. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 of 14 www.zamm-journal.org https://doi.org/10.1002/zamm.201900262