Olufemi OYADARE | Obafemi Awolowo University (original) (raw)

Papers by Olufemi OYADARE

The JEFT is the acronym for the Joint-Eigenspace Fourier Transform defined on a noncompact symmet... more The JEFT is the acronym for the Joint-Eigenspace Fourier Transform defined on a noncompact symmetric space. It is a consequence of a general construction of a Fourier transform modelled on the Harish-Chandra Fourier transform (on a semi-simple Lie group with finite centre) which (on the corresponding symmetric space of the noncompact type) serves as the Poisson-completion of the famous Helgason Fourier transform. For a noncompact semi-simple Lie group G (with finite centre) whose corresponding Lie algebra g has the Cartan decomposition g = t ⊕ p, its Iwasawa decomposition is given as G = KAN in which K is the analytic subgroup of G with Lie algebra t, A =: exp(a) (where a is a maximal abelian subspace of p) and N is the analytic subgroup of G corresponding to n = λ∈ + {X ∈ g : [H, X] = λ(H)X, ∀H ∈ a} (where + denote the set of all restricted positive roots). A member ϕ ∈ C(G), with ϕ(e) = 1 in which ϕ(k 1 gk 2) = ϕ(g), for all k 1 , k 2 ∈ K, g ∈ G is termed a spherical function and is said to belong to C(G//K). The Harish-Chandra spherical transform (on C(G//K)) written as f → f is defined on a * C as f (λ) = (f * ϕ λ)(e). Here * is the convolution on G, ϕ λ is the elementary spherical function corresponding to λ and e is the identity of G. It would be more satisfying to consider a general (Fourier) transform f → H g f of C(G) defined on a * C as (f * ϕ λ)(g), for every g ∈ G (not just for g = e of the above Harish-Chandra case) and to

We consider the irreducibility of the regular representation of a noncompact semisimpe Lie group ... more We consider the irreducibility of the regular representation of a noncompact semisimpe Lie group G on the Hilbert space of the image of the Joint-Eigenspace Fourier transform on its corresponding symmetric space G/K. The L 2 −decomposition of the Joint-Eigenspace Fourier transform leads to the complete characterization of the said irreducibility in terms of the simplicity of a pair of members of a * C .

arXiv (Cornell University), Jun 20, 2024

This paper develops the structure theory of a Malcev algebra via the consideration of its most im... more This paper develops the structure theory of a Malcev algebra via the consideration of its most important and largest Lie (sub-) algebra. We introduce the notion of a Lie algebra which uniquely corresponds to a Malcev algebra and use this correspondence to derive some basic properties of some types of ideals in the Malcev algebra. We then prove the exact nature of the root-space decomposition of a Malcev algebra.

This paper conducts a geometric analysis of the Joint-Eigenspace Fourier transform of the symmetr... more This paper conducts a geometric analysis of the Joint-Eigenspace Fourier transform of the symmetric space of the noncompact type. Our study shows how the Poisson transform builds up the well-known Helgason Fourier transform for an analysis of the complete duality of the underlying symetric space. Among other results, we establish an inversion formula, a Plancherel formula and (as our main result) a Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on any noncompact symmetric space.

We establish a K−type decomposition of the Harish-Chandra Schwartz algebra C p (G), for any real-... more We establish a K−type decomposition of the Harish-Chandra Schwartz algebra C p (G), for any real-rank 1 reductive group G with a maximal compact subgroup K and 0 < p ≤ 2. This decomposition is then used to give an infinite-matrix-realization of the operator-valued Fourier image F : C p (G) → C p (Ĝ) of C p (G) as a Fréchet multiplication algebra in which every member of C p (Ĝ) consists of a countable block-matrices of the form ((F B (α) (γ,m) (Λ) ⊗ F H (α) (γ,l) (Q : χ : ν)) γ∈F,(l,m)∈Z 2) F ⊂K,|F |<∞ for every α ∈ C p (G). This proves Trombi's conjecture for G of real rank 1 and the technique leads to a proof of the fundamental theorem of harmonic analysis for any arbitrary real-rank reductive group G.

International Journal of Algebra and Statistics, Nov 10, 2018

This paper presents a general method of constructing a complete set of Mutually Orthogonal Latin ... more This paper presents a general method of constructing a complete set of Mutually Orthogonal Latin Squares (MOLS) of the order of any prime, via the use of generating functions dened on the nite eld of this order. Apart from using the generating function to get a complete set of Mutually Orthogonal Latin Squares, the studies of the generating functions opens up the possibility of getting at the deep structural properties of MOLS. Copious examples were given for detailed illustrations.

International Journal of Algebra and Statistics, 2018

arXiv: Functional Analysis, Jun 21, 2017

This paper concerns the notion of a symmetric algebra and its generalization to a quasi-symmetric... more This paper concerns the notion of a symmetric algebra and its generalization to a quasi-symmetric algebra. We study the structure of these algebras in respect to their hull-kernel regularity and existence of some ideals, especially the hull-minimal ideals. §1. Introduction. It is well-known, from J. Ludwig (1998) that every semisimple symmetric polynomially bounded Fréchet algebra is hull-kernel regular and has a hull-minimal ideal generated by some elements of the algebra. This result gives a way of verifying the existence of hull-minimal ideals and of computing their basis elements in non-normable algebras of harmonic analysis, as has been shown for the Schwartz algebras of nilpotent and connected semisimple groups in J. Ludwig (1998) and O. O. Oyadare (2016) respectively. In this paper we give a generalization of the notion of a symmetric algebra to that of a quasi-symmetric and establish the importance of this generalization by showing that every semisimple quasi-symmetric polynomially bounded Fréchet algebra is hull-kernel regular. The results contained herein form a part of results of the author's thesis (O. O. Oyadare (2016)) at the University of Ibadan.

We introduce the notion of an admissible Mordell equation and establish some basic results concer... more We introduce the notion of an admissible Mordell equation and establish some basic results concerning properties of its integral solutions. A non class-number and complete parametrization, with the most minimal set of conditions, for generating all the integral solutions of admissible Mordell equations was then proved. An effective and efficient algorithm for computing the integral solutions of admissible Mordell equations was also established from a proof of the Hall conjecture. One of our major results concerns the upper-bound for the number of integral solutions of each of these equations which we show may be explicitly computed. The analysis of these results leads us to the problem of deriving a general expression for the number of integral solutions of the Mordell equations. This problem is completely solved.

This paper contains a new proof of Euler's theorem, that the only non-trivial integral soluti... more This paper contains a new proof of Euler's theorem, that the only non-trivial integral solution, (α, β), of α 2 = β 3 + 1 is (±3, 2). This proof employs only the properties of the ring, Z, of integers without recourse to elliptic curves and is independent of the methods of algebraic number fields. The advantage of our proof, over Euler's isolated and other known proofs of this result, is that it charts a common path to a novel approach to the solution of Catalan's conjecture and indeed of any Diophantine equation.

Journal of Generalized Lie Theory and Applications, 2016

This paper reconsiders the age-long problem of normed linear spaces which do not admit inner prod... more This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, F n (G), of real L p (G)−spaces (when G is a reductive group in the Harish-Chandra class and p=2n), the situation may be rectified, via an outlook which generalizes the fine structure of the Hilbert space, L 2 (G). This success opens the door for harmonic analysis of unitary representations, G→End(F n (G)), of G on the Hilbert-substructure F n (G), which has hitherto been considered impossible.

The central concept in the harmonic analysis of a compact group is the completeness of Peter-Weyl... more The central concept in the harmonic analysis of a compact group is the completeness of Peter-Weyl orthonormal basis as constructed from the matrix coefficients of a maximal set of irreducible unitary representations of the group, leading ultimately to the direct sum decomposition of its L^2- space. A Peter-Weyl theory for a semicomplete orthonormal set is also possible and is here developed in this paper for compact groups. Existence of semicomplete orthonormal sets on a compact group is proved by an explicit construction of the standard Riemann-Lebesgue semicomplete orthonormal set. This approach gives an insight into the role played by the L^2- space of a compact group, which is discovered to be just an example (indeed the largest example for every semicomplete orthonormal set) of what is called a prime-Parseval subspace, which we proved to be dense in the usual L^2- space, serves as the natural domain of the Fourier transform and breaks up into a direct-sum decomposition. This pa...

This paper contains a non-trivial generalization of the Harish-Chandra transforms on a connected ... more This paper contains a non-trivial generalization of the Harish-Chandra transforms on a connected semisimple Lie group G, with finite center, into what we term spherical convolutions. Among other results we show that its integral over the collection of bounded spherical functions at the identity element e ∈ G is a weighted Fourier transforms of the Abel transform at 0. Being a function on G, the restriction of this integral of its spherical Fourier transforms to the positive-definite spherical functions is then shown to be (the non-zero constant multiple of) a positive-definite distribution on G, which is tempered and invariant on G=SL(2,R). These results suggest the consideration of a calculus on the Schwartz algebras of spherical functions. The Plancherel measure of the spherical convolutions is also explicitly computed.

The major results of Barker [3.], leading to the spherical Bochner theorem and its (spherical) ex... more The major results of Barker [3.], leading to the spherical Bochner theorem and its (spherical) extension, were made possible through the spherical transform theory of Trombi-Varadarajan [14.] and were greatly controlled by the non-availability of the full (non-spherical) Harish-Chandra Fourier transform theory on a general connected semisimple Lie group, G. Sequel to the recently announced results of Oyadare [13.], where the full image of the Schwartz-type algebras, C^p(G), under the full Fourier transform is computed to be C^p(G):={(ξ_1)^-1· h· (ξ_1)^-1:h∈Z̅(F^ϵ)} with Z̅(F^ϵ) given as the Trombi-Varadarajan image of C^p(G//K), the present paper now gives the full Bochner theorem for G by lifting the results of [3.] to full non-spherical status. An extension of the full Bochner theorem to all of C^p(G),1≤ p≤2, is established. It is also conjectured that every positive-definite distribution T on G which corresponds to a Bochner measure μ on F^ϵ extends uniquely to an element of C^p(...

It is well-known that the Harish-Chandra transform, fHf, is a topological isomorphism of the sphe... more It is well-known that the Harish-Chandra transform, fHf, is a topological isomorphism of the spherical (Schwartz) convolution algebra C^p(G//K) (where K is a maximal compact subgroup of any arbitrarily chosen group G in the Harish-Chandra class and 0<p≤2) onto the (Schwartz) multiplication algebra Z̅(F^ϵ) (of w-invariant members of Z(F^ϵ), with ϵ=(2/p)-1). The same cannot however be said of the full Schwartz convolution algebra C^p(G), except for few specific examples of groups (notably G=SL(2,R)) and for some notable values of p (with restrictions on G and/or on C^p(G)). Nevertheless the full Harish-Chandra Plancherel formula on G is known for all of C^2(G)=:C(G). In order to then understand the structure of Harish-Chandra transform more clearly and to compute the image of C^p(G) under it (without any restriction) we derive an absolutely convergent series expansion (in terms of known functions) for the Harish-Chandra transform by an application of the full Plancherel formula on ...

This paper revisits the study of canonical wave-packets on a real reductive groups, sequel to its... more This paper revisits the study of canonical wave-packets on a real reductive groups, sequel to its construction in a recent manuscript of Oyadare. Among other results, we prove some decompositions of the Schwartz-type algebras using the canonical wave-packets.

arXiv (Cornell University), Jun 20, 2024

International Journal of Algebra and Statistics, Nov 10, 2018

International Journal of Algebra and Statistics, 2018

arXiv: Functional Analysis, Jun 21, 2017

Journal of Generalized Lie Theory and Applications, 2016

A study of the recently published Oyadare's image of the Schwartz-type algebras on SL(2,R).

This paper gives an explicit realization of the Fourier image of Harish-Chandra transform of a ge... more This paper gives an explicit realization of the Fourier image of Harish-Chandra transform of a general semisimple group, thus extending our earlier results to reductive groups which are not in Harish-Chandra class.

It is well-known that the Harish-Chandra transform, f → Hf, is a topological isomorphism of the s... more It is well-known that the Harish-Chandra transform, f → Hf, is a topological isomorphism of the spherical (Schwartz) convolution algebra C p (G//K) (where K is a maximal compact subgroup of any arbitrarily chosen group G in the Harish-Chandra class and 0 < p ≤ 2) onto the (Schwartz) multiplication algebra ¯ Z(F ϵ) (of w−invariant members of Z(F ϵ), with ϵ = (2/p) − 1). The same cannot however be said of the full Schwartz convolution algebra C p (G), except for few specific examples of groups (notably G = SL(2, R)) and for some notable values of p (with restrictions on G and/or on C p (G)). Nevertheless the full Harish-Chandra Plancherel formula on G is known for all of C 2 (G) =: C(G). In order to then understand the structure of Harish-Chandra transform more clearly and to compute the image of C p (G) under it (without any restriction) we derive an absolutely convergent series expansion (in terms of known functions) for the Harish-Chandra transform by an application of the full Plancherel formula on G. This leads to a computation of the image of C(G) under the Harish-Chandra transform which may be seen as a concrete realization of Arthur's result and be easily extended to all of C p (G) in much the same way as it is known in the work of Trombi and Varadarajan. 2010 Mathematics Subject Classification: 43A85, 22E30, 22E46

We give the exact contributions of Harish-Chandra transform, (Hf)(λ), of Schwartz functions f to ... more We give the exact contributions of Harish-Chandra transform, (Hf)(λ), of Schwartz functions f to the harmonic analysis of spherical convolutions and the corresponding L p − Schwartz algebras on a connected semisimple Lie group G (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of L p − Schwartz functions and the generalization of this Theorem under the full spherical convolution map.