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The analogue of the strong Szegö limit theorem on the 2- and 3-dimensional spheres. (English) Zbl 0866.47017
The author proves an analogue of the strong Szegö limit theorem on the 2- and 3-dimensional spheres. More precisely, let \(N=2\) or 3, and let \(f\) be a complex valued continuous function on \(S^N\) which is also in the Sobolev space of \(1/2\) order on \(S^N\). Assume that the closed convex hull of the image of \(f\) does not contain the origin, then \[ \begin{split} \text{trace}(\log(P_n\circ[f]\circ P_n))= d_n\int_{S^N}\log f(x)d\mu(x)\\ +\alpha_N\cdot n^{N-1}\int\int_{S^N\times S^N}{(\log f(x)-\log f(y))^2\over\sin^{N-1}d(x,y)\cdot\sin^2(d(x,y)/2)} d\mu(x)d\mu(y)+ o(n^{N-1})\end{split} \] as \(n\to\infty\). Here \(P_n\) denotes the orthogonal projection operator in \(L_2(S^N)\) onto the subspace \({\mathcal P}_n\) consisting of polynomials restricted to \(S^N\) of degree at most \(n\), \(d_n=\dim{\mathcal P}_n\), \([f]\) the multiplication operator by \(f\), and \(d(x,y)\) the geodesic distance between two points \(x\) and \(y\) on \(S^N\). Also \(d\mu(x)\) denotes the normalized volume element on \(S^N\), \(\alpha_2=1/(4\pi)\) and \(\alpha_3=1/(32)\). The author observes that in general the analogue of the strong Szegö limit theorem would be a formula giving the second-order asymptotics of \(\text{trace}(\log(P_n\circ[f]\circ P_n))\) with such coefficients that the first-order term is a linear functional of \(\log f\) and the second-order term is a bilinear functional of \(\log f\) for \(f\) in some class of functions for which the operator \(\log(P_n\circ[f]\circ P_n))\) is trace class.
Reviewer: K.Furutani

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
33C55 Spherical harmonics
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