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• W134832242 abstract "The Ph.D. thesis is mainly concerned with two generalizations of Szego limit theorem, in a “smooth” and in a “discontinuous” setting. The first generalization has common combinatorial background with certain problems for random walks. In the first chapter, we explicitly compute the third term in Szego asymptotic formula for Zoll manifolds. This includes the case of the operator of multiplication by a smooth function on the sphere in any dimension. A possible application is an expression for a regularized determinant of a multiplication operator, or a zeroth order pseudodifferential operator, on the two dimensional sphere. Further asymptotic terms can also be calculated by our method, however the corresponding computation becomes very involved. We also calculate moments and certain joint distributions for maximal non-negative excursions of random walks with independent identically distributed steps. The main tool in both proofs is a new combinatorial identity, which is called generalized Hunt–Dyson formula, abbreviated as gHD, and is equivalent to Bohnenblust–Spitzer theorem (BSt). In the second chapter, we prove gHD, which is a formula for the quantity ∑ σ∈S m [( max(0, xσ1 , xσ1 + xσ2 , · · · , xσ1 + · · · + xσm) )n − ( max(0, xσ1 , xσ1 + xσ2 , · · · , xσ1 + · · · + xσm−1) )n] , where the summation is taken over all permutations of m real variables x1, · · · , xm, and n is an arbitrary natural power. In the case of power n = 1 the latter identity reduces to the classical Hunt–Dyson formula (HD). After that we establish a connection between gHD and BSt. More precisely, we derive BSt from gHD and vice versa providing the former with a new proof. In the third chapter, a one term Szego type asymptotic formula with a sharp remainder estimate for a class of integral operators with symbols having discontinuities in both position variable and momentum is established. In this case a logarithmic factor appears in the asymptotics. 2000 Mathematics Subject Classification. Primary: 58J40, 58J37, 58J52, 47B35, 35Pxx, 35S05. Secondary: 60C05, 60G50, 05A19, 05E05. ISBN 91–7283–072–7 • TRITA-MAT–01–MA–03 • ISSN 1401–2278 • ISRN KTH/MAT/R–03/01–SE" @default.
• W134832242 created "2016-06-24" @default.
• W134832242 creator A5035243343 @default.
• W134832242 date "2001-01-01" @default.
• W134832242 modified "2023-09-24" @default.
• W134832242 title "Generalizations of Szego Limit Theorem : Higher Order Terms and Discontinuous Symbols" @default.
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• W134832242 hasPublicationYear "2001" @default.
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