Discrete spectra of convolutions of compactly supported functions on \(\mathrm{SE}(2)\) using Sturm-Liouville theory. (English) Zbl 1452.43002
Representations of functions \(f\in L^2(\text{SE}(2))\) with compact support are given, where \(\text{SE}(2)\) is the semidirect product of \(\mathbb{R}^2\) with the special orthogonal group \(\text{SO}(2)\). With \(B_a^2=\{x\in \mathbb{R}^2:\|x\|\le a\}\), \(a>0\), the set \(D_a^2=B_a^2\times\text{SO}(2)\) is a compact subset of \(\text{SE}(2)\). The series expansions are with respect to orthogonal bases of \(L_2(D_a^2)\), which involve (generalized) Bessel functions as eigenfunctions of the Dirichlet and Neumann problems for the (generalized) Bessel differential equation on \((0,a)\), respectively. Explicit integral representations for the Fourier coefficients of functions and of convolutions of functions are given.
Reviewer: Manfred Möller (Johannesburg)
MSC:
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 34B24 | Sturm-Liouville theory |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
| 43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
Keywords:
orthogonal \(L_2\) basis on two-dimensional special Euclidean group; series expansion; series expansion of convolutions; Dirichlet boundary condition; Neumann boundary condition; Plancherel formulaReferences:
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