Relations between the norms of a function and its gradient in classes of surface and solid spherical harmonics on a finite-dimensional space

Relations between the Euclidean (mean-square) and Chebyshev (uniform) norms of a surface spherical harmonic of order n and its gradient in the tangent bundle of the sphere \(\mathbb{S}^{k - 1} \), as well as relations between the same norms of a solid spherical harmonic (internal and external) of order n and its spatial gradient, are given. In all cases, the norm of the gradient differs from that of the harmonic by a factor of order n. As examples, relations between the norms of the internal and external solid spherical harmonics

$r\left\langle {gradU_n } \right\rangle = n\left\langle {U_n } \right\rangle , r\left\| {gradU_n } \right\| = \sqrt {n(2n + k - 2)} \left\| {U_n } \right\|$

and

$r\left\langle {gradV_n } \right\rangle = (n + k - 2)\left\langle {V_n } \right\rangle , r\left\| {gradV_n } \right\| = \sqrt {(n + k - 2)(2n + k - 2)} \left\| {V_n } \right\|,$

respectively, are considered; here, <·> and ∥·∥ denote the Chebyshev and Euclidean norms, respectively.