Fourier inversion for multidimensional characteristic functions (Q1209215)

Let \(f\) be a density function on \(\mathbb{R}^ n\), whose mean value function (i.e. \(r\in(0,+\infty)\to M_ rf(x)=\int_{S^{n- 1}}f(x+r\omega)d\omega\), where \(d\omega\) is normalized surface measure on the unit sphere) is piecewise \(C^ \infty\) and compactly supported. The characteristic function and its spherical partial sum are respectively defined by: \[ \hat f(\mu)=\int_{\mathbb{R}^ n}f(x)e^{i\langle\mu,x\rangle}dx,\qquad f_ R(x)={1\over (2\pi)^ n}\int_{|\mu|\leq R}\hat f(\mu)e^{- i\langle\mu,x\rangle}d\mu. \] Given a function \(f\) as above, some asymptotic results are proved; more precisely it is stated that \(\lim_{R\to+\infty}f_ R(x)=M_{0+}f(x)\) if and only if \(M_ rf\) has \(k=[(n-3)/2]\) continuous derivatives; otherwise \[ \liminf_{R\to+\infty}R^{-\nu}[f_ R(x)- M_{0+}f(x)]<0<\limsup_{R\to+\infty} R^{-\nu}[f_ R(x)- M_{0+}f(x)], \] where \(\nu\) \((\geq0)\) is uniquely determined. To achieve this result, some properties and asymptotic formulas for the spherical Bessel function and the Dirichlet kernel are proved. In conclusion some examples are discussed and some remarks concerning the Gibbs phenomenon are made.