  The spectral theory on the S-spectrum was introduced to give an appropriate
mathematical setting to quaternionic quantum mechanics, but it was soon
realized that there were different applications of this theory, for example, to
fractional heat diffusion and to the spectral theory for the Dirac operator on
manifolds. In this seminal paper we introduce the harmonic functional calculus
based on the S-spectrum and on an integral representation of axially harmonic
functions. This calculus can be seen as a bridge between harmonic analysis and
the spectral theory. The resolvent operator of the harmonic functional calculus
is the commutative version of the pseudo S-resolvent operator. This new
calculus also appears, in a natural way, in the product rule for the
F-functional calculus.
