108 research outputs found
Generalizing the arithmatic geometric mean — a hapless computer experiment
The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ”hapless computer experiment” in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ”fluctuations” are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt. The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ”hapless computer experiment” in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ”fluctuations” are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt
Fourier analysis of a space of Hilbert-Schmidt operators-new Ha-plitz type operators
If a group acts via unitary operators on a Hilbert space of functions then this group action extends in an obvious way to the space of Hilbert- Schmidt operators over the given Hilbert space. Even if the action on functions is irreducible, the action on H.S . operators need not be irreducible. It is often of considerable interest to find out what the irreducible constituents are. Such an attitude has recently been advocated in the theory of "Ha-plitz" (Hankel + Toeplitz) operators. In this paper we solve this problem the space of H.S . operators over the Hilbert space L2(Δ,πα) of square integrable functions over the unit disk Δ equipped with the Dzhrbashyan measure dμ(z) = (α+1)(1- z)αdA(z)(α > -1). This complements the earlier results. In particular we discover many new Ha-plitz type operators. The question of their smoothness properties (Sp- estimates etc.) is however only touched upon
The Bellaterra connection
A new object is introduced - the "Fischer bundle" . It is, formally speaking, an Hermitean bundle of infinite rank over a bounded symmetric domain whose fibers are Hilbert spaces whose elements can be realized as entire analytic functions square integrable with respect to a Gaussian measure ("Fischer spaces" ). The definition was inspired by our previous work on the "Fock bundle" . An even more general framework is indicated, which allows one to look upon the two concepts from a unified point of view
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