Precise knowledge of magnetic fields is crucial in many medical imaging applications such as magnetic resonance imaging (MRI) or magnetic particle imaging (MPI), as they form the foundation of these imaging systems. Mathematical methods are essential for efficiently analysing the magnetic fields in the entire field-of-view. In this work, we propose a compact and unique representation of the magnetic fields using real solid spherical harmonic expansions, which can be obtained by spherical t-designs. To ensure a unique representation, the expansion point is shifted at the level of the expansion coefficients. As an application scenario, these methods are used to acquire and analyse the magnetic fields of an MPI system. Here, the field-free-point of the spatial encoding field serves as the unique expansion point.

Let $D$ be the $n$-dimensional Lie ball and let $B\text{(S)}$ be the space of hyperfunctions on the Shilov boundary $S$ of $D$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform ${{P}_{l,\text{ }\!\!\lambda\!\!\text{ }}}f$ of an element $f$ in the space $B\text{(S)}$ for $f$ to be in ${{L}^{p}}\left( S \right)$, $1\,<\,p\,<\,\infty $. Namely, if $F$ is the Poisson transform of some $f\in \,B(S)$$F\,=\,{{P}_{l,\lambda }}f$), then for any $l\,\in \,Z$) and $\lambda \,\in \,C$ such that $Re[\text{i}\lambda ] > \frac{n}{2}\,-\,1$, we show that $f\,\in \,{{L}^{p}}\text{(}S\text{)}$ if and only if $f$ satisfies the growth condition

$${{\left\| F \right\|}_{\lambda ,p}}=\underset{0\le r<1}{\mathop{\sup }}\,{{\left( 1\,-\,{{r}^{2}} \right)}^{\operatorname{Re}\left[ \text{i }\lambda \text{ } \right]-\frac{n}{2}+l}}{{\left[ \,\int_{s}{{{\left| F\left( ru \right) \right|}^{p}}du} \right]}^{\frac{1}{p}}}<\,+\infty $$

For a convex body $K$ in $\R^d$ and $1\le k\le d-1$, let $P_k(K)$ be the Minkowski sum (average) of all orthogonal projections of $K$ onto $k$-dimensional subspaces of $\R^d$. It is known that the operator $P_k$ is injective if $k\,{\geq}\, d/2$, $k\,{=}\,3$ for all $d$, and if $k=2$, $d\ne 14$.

It is shown that $P_{2k}(K)$ determines a convex body $K$ among all centrally symmetric convex bodies and $P_{2k+1}(K)$ determines a convex body $K$ among all bodies of constant width. Corresponding stability results are also given. Furthermore, it is shown that any convex body $K$ is determined by the two sets $P_k(K)$ and $P_{k'}(K)$ if $1<k<k'$. Concerning the range of $P_k$, $1\le k\le d-2$, it is shown that its closure (in the Hausdorff-metric) does not contain any polytopes other than singletons.

In this paper we estimate the $\left( {{L}^{p}}-{{L}^{2}} \right)$-norm of the complex harmonic projectors $\pi \ell {\ell }',\,1\le p\le 2$, uniformly with respect to the indexes $\ell,{\ell}'$. We provide sharp estimates both for the projectors ${{\pi }_{\ell{\ell}'}}$, when $\ell,{\ell}'$ belong to a proper angular sector in $\mathbb{N}\,\times \,\mathbb{N}$, and for the projectors ${{\pi }_{\ell0}}$ and ${{\pi }_{0\ell}}$. The proof is based on an extension of a complex interpolation argument by C. Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the ${{L}^{1}}$ norm on the unit sphere of ${{\mathbb{R}}^{2n}}$.

An integral formula is derived for the spherical functions on the symmetric space ${G}/{K\,=\,{\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;}\;$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra a of the abelian part in the decomposition $G\,=\,KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space ${\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;$ using the Plancherel formula.

In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.