CHAPTER 4.8 - Fourier Series and Spherical Harmonics in Convexity

Abstract

This chapter discusses major geometric results obtained by the use of Fourier series or spherical harmonics. The isoperimetric inequality for plane domains can be deduced from simple properties of Fourier series. Many applications of spherical harmonics to problems concerning geometric inequalities depend on the Laplace–Behrami operator. The analog of the Laplace operator for functions on Sd–1 is the Laplace–Beltrami operator, which is denoted by Δo. A homogeneous polynomial p(x) in d variables with real coefficients is called a harmonic polynomial if Δp(x) = 0. The restriction of a harmonic polynomial of degree n to Sd–1 is called a spherical harmonic of order n and dimension d. The spherical harmonics are eigenfunctions of the Laplace–Beltrami operator. For geometric applications, a very useful result for spherical harmonics is the Funk–Hecke theorem. The chapter focuses mainly on theorems that have no natural extension to the d-dimensional situation and discusses results that provide good illustrations of the general methods under the simplifying assumption that d = 2.