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By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on the constant function equal to one. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expressions for zonal harmonics starting from the 2 and 3 dimensional cases. It turns out that all zonal harmonics in any dimension are related to the real part of powers of the standard Hermitian product in $\mathbb{C}$. At the end we compare formulas, obtaining interesting equalities involving the real part of positive and negative powers of the standard Hermitian product. In the two appendices we show how our computations are optimal compared to direct ones., Comment: 27 Pages; 2 Appendices; Comments are welcome!

We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with polyharmonic functions, the elements of the kernel of an iterated Laplacian. Here we consider polynomials of the form $P(x)=\sum_{k=0}^d x^ka_k$, with Clifford coefficients $a_k\in\mathbb R_{n}$, and get an analogous decomposition related to zonal polyharmonics. We show the relation between such decomposition and the Dirac (or Cauchy-Riemann) operator and extend the results to slice-regular functions., Comment: to be published in Complex variables and Elliptic Equations. arXiv admin note: text overlap with arXiv:1801.03045

The Goursat representation formula in the complex plane, expressing a real-valued biharmonic function in terms of two holomorphic functions and their anti-holomorphic complex conjugates, is generalized to Euclidean space, expressing a real-valued polyharmonic function of order p in terms of p the so called monogenic functions of Clifford analysis.

In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.

We consider the problem of interpolation to a sequence of n-variate periodic data functions prescribed on {j}×Rn, j∈Z+, from a space of piecewise polyharmonic functions (polysplines) of n+1 variables. A unique solution is obtained subject to boundary conditions of the type employed in Duchon’s theory of polyharmonic surface splines. The construction of the polyspline scheme is based on the extension of Schoenberg’s semi-cardinal interpolation model to a class of univariate L-splines.

We study planar complex-valued functions that satisfy a certain Wirtinger differential equation of order k. Our considerations include entire functions (k=1), harmonic functions (k=2), biharmonic functions (k=4), and polyharmonic functions (k even) in general. Under the assumption of restricted exponential growth and square integrability along the real axis, we establish a sampling theorem that extends the classical sampling theorem of Whittaker–Kotel’nikov–Shannon and reduces to the latter when k=1. Intermediate steps, which may be of independent interest, are representation theorems, uniqueness theorems, and the construction of fundamental functions for interpolation. We also consider supplements, variants, generalizations, and an algorithm.

We show that the spaces of harmonic functions with respect to the Poincare metric in the unit ball BN in R N have many different properties depending upon whether N is even or odd.

A system of functions 0-normalized with respect to the operator Δ in some domain Ω⊂Rn is constructed. Application of this system to boundary value problems for the polyharmonic equation is considered. Connection between harmonic functions and solutions of the Helmholtz equation is investigated.

We show that the scaling spaces defined by the polysplines of order p p provide approximation order 2 p . 2p. For that purpose we refine the results on one-dimensional approximation order by L L -splines obtained by de Boor, DeVore, and Ron (1994).

In this note we prove the following theorem:Suppose 0−1. Then there is a constant C=C(p,m,n,α) such that ∫Bu(x)p1−|x|αdV(x)⩽Cu(0)p+∫B∇u(x)p1−|x|p+αdV(x), for all polyharmonic functions u of order m, on the unit ball B⊂Rn.

Polyharmonic functions f of infinite order and type {\tau} on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of infinite order and type 0 can be extended to analytic functions on the complex plane cut along the negative semiaxis. The second main result gives a constructive procedure via Fourier-Laplace series for the analytic extension of a polyharmonic function on annular region A(r_{0},r_{1}) of infinite order and type less than 1/2r_{1} to the kernel of the harmonicity hull of the annular region. The methods of proof depend on an extensive investigation of Taylor series with respect to linear differential operators with constant coefficients., Comment: 32 pages

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C ∞ or analytic functions in the ball B R . We prove two main results on the interpolation of C ∞ or analytic functions f in the ball B R by polyharmonic functions h of a given order of polyharmonicity p.

The aim of this note is to generalize the Liouville theorem to polyharmonic functions $u$ on $R^{n}$ . We give a condition on spherical means to assure that $u$ is a polynomial.

In this article, we consider a class of Dirichlet problems with Lp boundary data for polyharmonic functions in the upper-half space. By introducing a sequence of new kernel functions for the upper-half space, called higher order Poisson kernels, integral representation solutions of the problems are provided.

We obtain in explicit form the unique Gaussian cubature for balls (spheres) in Rn based on integrals over balls (spheres), centered at the origin, that integrates exactly all m-harmonic functions. In particular, this formula is exact for all polynomials in n variables of degree 2m−1. A Gaussian cubature for simplices is also constructed. Upper bounds for the errors for certain smoothness classes are derived.