On positive values of spherical harmonics and trigonometric polynomials (Q1889629)

Let \(x=(x_1,\dots,x_n)\in \mathbb R^n \) and \(S^{n-1}=\{ x\in \mathbb R^n:| x| =1 \} \) be the unit sphere. Further, consider the norm \[ \|f \|_p=\left \{ \int _{S^{n-1}}| f(x)| ^p \,dx\right \} ^{1/p}, \quad p\geq 1, \] on the space \(L_p(f)\). If \(f(x)\) belongs to the set of \(H_k^n-\)homogeneous harmonic polynomials of order \(k\) then the restriction of \(f(x)\) to \(S^{n-1}\) is called a spherical harmonic \(Y_k(x)\) of order \(k\). Problems of one-side approximation, recently considered by \textit{D. H. Armitage, S. J. Gardiner} [Proceedings of the 3rd international conference on multivariate approximation theory, Witten-Bommerholz, Germany, 1998, W.Haussmann et Al. editors , 43--56 (1999; Zbl 0973.41012)], led to the question of the determination of \(Y_k\) with least measure of its positivity \(P(Y_k)=\{ x\in S^{n-1}:Y_k(x) \geq 0\} \). The positivity is meaningful for even \(k\) and \(n\geq 3\). In this paper, the author presents only order estimates of measure \(P(Y_k)\gg 1/\sqrt k,\), if \(n=3\) and measure \(P(Y_k)\gg 1/k\) if \(n=4\). A similar question on the multidimensional torus \(\mathbb T^n =\mathbb R^n /\mathbb Z^n \) is considered.