An Orthonormal System of Modified Spherical Harmonics

The motivation for the theory of modified spherical harmonics is the desire to find a class of real functions on the half-sphere S+=(x,y,t):x2+y2+t2=1,t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{+}=\left\{ (x,y,t): x^2 + y^2 + t^2 =1,\right. \left. t > 0 \right\} $$\end{document} in R3={(x,y,t)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3 = \{(x,y,t)\}$$\end{document} which is naturally adapted to S+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{+}$$\end{document} in the sense that it behaves like the classical spherical harmonics on the full unit sphere. Such a system is at hand if we replace the Laplace equation Δh=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta h =0$$\end{document} by the equation tΔv+∂v∂t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\Delta v+\frac{\partial v}{\partial t}=0$$\end{document} and consider the restrictions of the polynomial solutions v=v(x,y,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v=v(x,y,t)$$\end{document} to the half-sphere S+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{+}$$\end{document}. Elements of this class of functions are called “modified spherical harmonics”. In order to get similar results as in case of the classical spherical harmonics one has however to replace the Euclidean scalar product on the unit sphere in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document} by a non-Euclidean one defined on S+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{+}$$\end{document}. All this has already been worked out in an earlier paper entitled “Modified Spherical Harmonics”. In the present paper we deduct an explicit orthonormal system of modified spherical harmonics. Such a system was missing.