Fourier coefficients and growth of harmonic functions

Abstract

We consider Harmonic Functions, H of several variables. We obtain necessary and sufficient conditions on its Fourier coefficients so that H is an entire harmonic (that is, has no finite singularities) function; the radius of harmonicity in terms of its Fourier coefficients in case H is not entire. Further, we obtain, in terms of its Fourier coefficients, the Order and Type growth measures, both in case H is entire or non-entire.

Keywords

• harmonic function
• spherical harmonics
• entire harmonic function
• radius of harmonicity
• Fourier coefficients
• order and type growth measures
• Liouville's theorem.