Fix real numbers α ≥ β ≥ - 1/2, with α > - 1/2, and equip [-1, 1] with the measure dμ(x) = (1 - x)α(1 + x)β dx. For p = 4(α + l)/(2α + 3) there exists f ∈ Lp(μ) such that f(x) = 0 a.e. on [-1, 0] and the appropriate Jacobi polynomial series for f diverges a.e. on [-1.1] This implies failure of localization for spherical harmonic expansions of elements of L2d/(d+1)(X), where X is a sphere or projective space of dimension d > 1.
- Compact two-point homogeneous space
- Jacobi polynomial
- Laplace-Beltrami operator
- Spherical harmonic