Low-degree spline quasi-interpolants in the Bernstein basis

•The referees’ comments have been incorporated into the new version of the manuscript, correcting many typos.•The structure of the paper has been improved and the number of figures has been reduced.•Some maintained graphs have been combined to reduce the number of figures.•Some tables have been combined for the same purpose.•The notation used for the quasi-interpolation operators has been improved.•In the quadratic case, a very interesting comment by reviewer 3 has been incorporated, which relates the quasi-interpolation scheme introduced in the article to a quasi-interpolant obtained by discretisation of the linear functional of the classical quadratic differential quasi-interpolant.•The constructive method introduced in the article has the potential to give rise to new quasi-interpolants that incorporate specific properties.