Richard AWONUSIKA - Academia.edu (original) (raw)
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This thesis is concerned with the spectral theory of the Laplacian on non-Euclidean spaces and it... more This thesis is concerned with the spectral theory of the Laplacian on non-Euclidean spaces and its intimate links with harmonic analysis and the theory of special functions. More specifically, it studies the spectral theory of the Laplacian on the quotients M = Γ \G/K and X = G/K, where G is a connected semisimple Lie group, K is a maximal compact subgroup of G and Γ is a discrete subgroup of G. It builds upon the special cases of compact and noncompact hyperbolic surfaces M = Γ \H where Γ ⊂ P SL(2, R) is a Fuchsian group and G := P SL(2, R) is the projective special linear group of all 2×2 real matrices with determinant 1 (the latter is the group of orientation-preserving isometries of the Poincaré upper half-plane H = {z ∈ C : Im z > 0} and by a Fuchsian group we mean a discrete cofinite subgroup of P SL(2, R) that acts properly discontinuously on H) and it generalises these techniques to explicit constructions of various spectral functions on n-dimensional symmetric spaces X , most notably, when X = the unit sphere S n ; the Euclidean space R n ; the real projective space RP n ; the complex projective space CP n ; the hyperbolic upper half-space H n ; and the hyperbolic unit ball D n. The main tools in contemplating this throughout are the use of various spectral estimates and identities, integral representations and the indispensable trace formulae. v far. I would also want to thank all of my friends who supported me in one way or the other and encouraged me to strive towards my goal. This acknowledgement will be incomplete if I seize to register my sincere appreciation to both the academic and non-academic staff of the Department of Mathematical Sciences, Adekunle Ajasin University, Akungba-Akoko, for their assistance in ensuring that I was properly and periodically updated about the happenings in this great citadel of learning.
Canadian Mathematical Bulletin, 2017
The Jacobi coefficientsare linked to the Maclaurin spectral expansion of the Schwartz kernel of f... more The Jacobi coefficientsare linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the Jacobi polynomialsinto a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed, and a direct trace interpretation of the Maclaurin coefficients is presented.
This thesis is concerned with the spectral theory of the Laplacian on non-Euclidean spaces and it... more This thesis is concerned with the spectral theory of the Laplacian on non-Euclidean spaces and its intimate links with harmonic analysis and the theory of special functions. More specifically, it studies the spectral theory of the Laplacian on the quotients M = Γ \G/K and X = G/K, where G is a connected semisimple Lie group, K is a maximal compact subgroup of G and Γ is a discrete subgroup of G. It builds upon the special cases of compact and noncompact hyperbolic surfaces M = Γ \H where Γ ⊂ P SL(2, R) is a Fuchsian group and G := P SL(2, R) is the projective special linear group of all 2×2 real matrices with determinant 1 (the latter is the group of orientation-preserving isometries of the Poincaré upper half-plane H = {z ∈ C : Im z > 0} and by a Fuchsian group we mean a discrete cofinite subgroup of P SL(2, R) that acts properly discontinuously on H) and it generalises these techniques to explicit constructions of various spectral functions on n-dimensional symmetric spaces X , most notably, when X = the unit sphere S n ; the Euclidean space R n ; the real projective space RP n ; the complex projective space CP n ; the hyperbolic upper half-space H n ; and the hyperbolic unit ball D n. The main tools in contemplating this throughout are the use of various spectral estimates and identities, integral representations and the indispensable trace formulae. v far. I would also want to thank all of my friends who supported me in one way or the other and encouraged me to strive towards my goal. This acknowledgement will be incomplete if I seize to register my sincere appreciation to both the academic and non-academic staff of the Department of Mathematical Sciences, Adekunle Ajasin University, Akungba-Akoko, for their assistance in ensuring that I was properly and periodically updated about the happenings in this great citadel of learning.
Canadian Mathematical Bulletin, 2017
The Jacobi coefficientsare linked to the Maclaurin spectral expansion of the Schwartz kernel of f... more The Jacobi coefficientsare linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the Jacobi polynomialsinto a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed, and a direct trace interpretation of the Maclaurin coefficients is presented.