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.

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.

The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, i.e., null solutions of the generalized Cauchy-Riemann operator in R 4 , denoted by D. This theorem is divided in two steps. In the first step a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for these type of functions is the starting point of the S-functional calculus. In the second step a monogenic function is obtained by applying the Laplace operator in four real variables, namely ∆, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of ∆ = DD. Instead of applying directly the Laplace operator to a slice hyperholomorphic function we apply first the operator D and we get a polyanalytic function of order 2, i..e, a function that belongs to the kernel of D 2 . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S-spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

We study the nonlinear almost compressible 2D Oberbeck-Boussinesq system, characterized by an extra buoyancy term where the density depends on the pressure, and a corresponding dimensionless parameter β, proportional to the (positive) compressibility factor β 0. The local in time existence of the perturbation to the conductive solution is proved for any "size" of the initial data. However, unlike the classical problem where β 0 = 0, a smallness condition on the initial data is needed for global in time existence, along with smallness of the Rayleigh number. Removing this condition appears quite challenging and we leave it as an open question.

In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.