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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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