Harmonic analysis on spheres, I

Harmonic analysis on spheres

1. Calculus on spheres 2. Spherical Laplacian from Euclidean 3. Eigenvectors for the spherical Laplacian 4. Invariant integrals on spheres 5. L spectral decompositions on spheres 6. Sup-norms of spherical harmonics on Sn−1 7. Pointwise convergence of Fourier-Laplace series 8. Irreducibility of representation spaces for O(n) 9. Hecke’s identity • Appendix: Bernstein’s proof of Weierstraß approxi...

Harmonic analysis on spheres, II

1. A doomed-but-plausible trial calculation 2. Matrix exponentiation 3. Rotation-invariant derivatives on R 4. Instrinsic derivatives 5. Aside: GLn(R)-invariance of the Euler operator 6. ∑ i<j X 2 ij in coordinates on R 7. X` commutes with ∑ i<j X 2 ij 8. Stabilization of Hd by Xγ 9. Integration by parts for Xγ 10. Sup norms of derivatives of harmonic polynomials 11. Termwise differentiation of...

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

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.