  We design quasi-interpolation operators based on piecewise polynomial weight
functions of degree less than or equal to $p$ that map into the space of
continuous piecewise polynomials of degree less than or equal to $p+1$. We show
that the operators have optimal approximation properties, i.e., of order $p+2$.
This can be exploited to enhance the accuracy of finite element approximations
provided that they are sufficiently close to the orthogonal projection of the
exact solution on the space of piecewise polynomials of degree less than or
equal to $p$. Such a condition is met by various numerical schemes, e.g., mixed
finite element methods and discontinuous Petrov--Galerkin methods. Contrary to
well-established postprocessing techniques which also require this or a similar
closeness property, our proposed method delivers a conforming postprocessed
solution that does not rely on discrete approximations of derivatives nor local
versions of the underlying PDE. In addition, we introduce a second family of
quasi-interpolation operators that are based on piecewise constant weight
functions, which can be used, e.g., to postprocess solutions of hybridizable
discontinuous Galerkin methods. Another application of our proposed operators
is the definition of projection operators bounded in Sobolev spaces with
negative indices. Numerical examples demonstrate the effectiveness of our
approach.
