# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] R. Ya. Doktorskiĭ, Generalization of the Szegő limit theorem to the multidimensional case, Sibirsk. Mat. Zh. 25 (1984), no. 5, 20 – 29 (Russian). [2] B. L. Golinskiĭ and I. A. Ibragimov, A limit theorm of G. Szegő, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 408 – 427 (Russian). [3] Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. · Zbl 0080.09501 [4] I. I. Hirschman Jr., The strong Szegö limit theorem for Toeplitz determinants, Amer. J. Math. 88 (1966), 577 – 614. · Zbl 0173.42602 [5] Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. Lars Hörmander, The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. Fourier integral operators. [6] K. Johansson, On Szegö’s asymptotic formula for Toeplitz determinants and generalizations, Phd Diss., UUDM Report 15, Uppsala Univ. Dept. of Math., Uppsala, Sweden, 1987. · Zbl 0661.30001 [7] M. Kac, Toeplitz matrices, translation kernels and a related problem in probability theory, Duke Math. J. 21 (1954), 501–509. · Zbl 0056.10201 [8] A. Laptev and Yu. Safarov, Error estimate in the generalized Szegő theorem, Journées ”Équations aux Dérivées Partielles” (Saint Jean de Monts, 1991) École Polytech., Palaiseau, 1991, pp. Exp. No. XV, 7. · Zbl 0743.58032 [9] I. Ju. Linnik, A multidimensional analogue of G. Szegő’s limit theorem, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 6, 1393 – 1403, 1439 (Russian). [10] K. Okikiolu, The analogue of the strong Szegö limit theorem on the torus and the 3-sphere, Ph.d. Diss., Dept. Math UCLA, Los Angeles, California, (1991). [11] K. Okikiolu, The analogue of the strong Szegö limit theorem for a Sturm-Liouville operator on the interval, preprint, (1992). [12] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. · Zbl 0232.42007 [13] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. · JFM 65.0278.03 [14] Alejandro Uribe, A symbol calculus for a class of pseudodifferential operators on \?$$^{n}$$ and band asymptotics, J. Funct. Anal. 59 (1984), no. 3, 535 – 556. · Zbl 0561.35082 [15] N. Ja. Vilenkin, Special functions and the theory of group representations, Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. [16] Harold Widom, Asymptotic expansions for pseudodifferential operators on bounded domains, Lecture Notes in Mathematics, vol. 1152, Springer-Verlag, Berlin, 1985. · Zbl 0582.35002 [17] Harold Widom, Szegő’s theorem and a complete symbolic calculus for pseudodifferential operators, Seminar on Singularities of Solutions of Linear Partial Differential Equations (Inst. Adv. Study, Princeton, N.J., 1977/78) Ann. of Math. Stud., vol. 91, Princeton Univ. Press, Princeton, N.J., 1979, pp. 261 – 283.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.