  Any positive power of the Laplacian is related via its Fourier symbol to a
hypersingular integral with finite differences. We show how this yields a
pointwise evaluation which is more flexible than other notions used so far in
the literature for powers larger than 1; in particular, this evaluation can be
applied to more general boundary value problems and we exhibit explicit
examples. We also provide a natural variational framework and, using an
asymptotic analysis, we prove how these hypersingular integrals reduce to
polyharmonic operators in some cases. Our presentation aims to be as
self-contained as possible and relies on elementary pointwise calculations and
known identities for special functions.
