Harmonic analysis on spheres

Twistor Interpretation of Harmonic Spheres and Yang–Mills Fields

We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space ΩG of a compact Lie group G a...

Approximation Theory and Harmonic Analysis on Spheres and Balls

Harmonic analysis on spheres, I

Harmonic analysis on the line is the theory of Fourier transforms, more complicated than Fourier series, due to the line’s non-compactness. On R the exponential functions, while still eigenfunctions for d dx and still giving group homomorphisms, are no longer in L(R). Entangled with this point is the fact that Fourier inversion expresses functions as integrals of exponential functions, not as s...

ar X iv : d g - ga / 9 51 20 10 v 2 2 4 M ay 1 99 6 Weierstrass representations for harmonic morphisms on Euclidean spaces and spheres

We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic mor-phisms from Euclidean spaces and spheres.

On continuum incidence problems related to harmonic analysis

The purpose of this note is to put some recent work in harmonic analysis involving combinatorics of circles and spheres in perspective. A review of some of this work was given by Wolff in [17], and we will not duplicate what is said there. On the other hand, Wolff’s recent paper [18] allows for some simplifications beyond [17]. More precisely, the difficult argument in [16] can be considerably ...