This paper deals with Hermite osculatory interpolating splines. For a partition of a real interval endowed with a refinement consisting in dividing each subinterval into two small subintervals, we consider a space of smooth splines with additional smoothness at the vertices of the initial partition, and of the lowest possible degree. A normalized B-spline-like representation for the considered spline space is provided. In addition, several quasi-interpolation operators based on blossoming and control polynomials have also been developed. Some numerical tests are presented and compared with some recent works to illustrate the performance of the proposed approach.

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