Convergence rate of spherical harmonic expansions of smooth functions (Q947537)

Let \(S^{d-1}= \{x\in\mathbb{R}^d:|x|=1\}\) be the unit sphere in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) and let \(L^p(s^{d-1})\), \(0< p<\infty\) denote the Lebesgue space on \(S^{d-1}\). Let \(Y_k(f)\) denote the orthogonal projection of \(f\) onto the space of spherio harmonics of degree \(k\). Given a nonnegative function \(\varphi\) on \([0,\infty)\) we define the \(\varphi\)-derivative \(f^{(\varphi)}\circ f\in L(s^{d-1})\) in a distributional sense by \(Y_k(\xi^{(\varphi)})= \varphi(x)Y_k(f)\), \(k= 0,1,2,\dots\). For \(\delta> -1\), \((C,\delta)\) operator of \(f\sim \sum^\infty_{k=0} Y_k(f)\) is defined as \[ \sigma^\delta_N(f)= {1\over A^\delta_N} \sum^N_{k=0} A^\delta_{n-k} Y_k(f),\quad N= 0,1,2,\dots\;. \] We define the minimal \((c,\delta)\) operators as \(\sigma^\delta_x(f)(x)= \sup_N|\sigma^\delta_N(f)(x)|\). In this paper the authors study the rate of convergence of spherical harmonic expansions of smooth functions. Their main result is as follows: Theorem. Assume that \(\delta> 0\), \(\ell\) is a positive integer and \(\varphi\in C^{p+1}[0,\infty)\) with \(\varphi(0)= 0\), \(\lim_{t\to\infty} \varphi(p)=\infty\) and \(\varphi(t)> 0\) for \(t\in (0,\infty)\). Then for \(f\in L^1_\varphi(s^{d-1})\) we have for a.e. \(x\in s^{d-1}\) \[ |f(x)- \sigma_N(f)(x)|\in C\Biggl[\int^\infty_1\Biggl| \mu^{(p+1)}(t){t^{p+1}\over t+ N}\Biggr|\, dt\Biggr]\sigma^\alpha_k(f^{\varphi)})(x), \] where \(\alpha= \min\{\delta,\ell\}\), \(\mu(t)= {1\over\varphi(p)}\) and \(c\) is a constant independent of \(N\), \(x\) and \(f\).