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. Efficient 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.

中文翻译：

---

关于代数三角整数样条

在本文中，我们提出了一种代数和三角函数的二次逼近算子。它被称为整数二次样条插值，它与在相同间隔上的单变量实值函数的给定积分值一致，而不是与节点处的函数值一致。构造并充分研究了基于该插值的分数积分和分数Caputo导数的有效近似。还研究了一般的近似误差，并且当间隔被等分时，还获得了超收敛性。数值算例表明，该方法是有效的，并且二次代数三角积分样条具有更高的逼近能力。