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Fourier-Bessel series of compactly supported convolutions on disks. (English) Zbl 07528660

The authors obtain fundamental properties for Fourier-Bessel coefficients of convolutions of functions supported on disks. More precisely, for \(s\geq 0\) and \(m\in\mathbb{Z}\), let us consider the polar harmonics defined by \(\Phi^m_s(ru_\theta)=J_m(sr)\mathcal{Y}_m(\theta)\), where \(\mathcal{Y}_m(\theta)=\frac{e^{im\theta}}{\sqrt{2\pi}}\) and \(J_m(x)\) is the \(m\)th-order Bessel function of the first kind. For every \(m\in\mathbb{Z}\) let \(\{z_{m,n}\}_{n\in\mathbb{N}}\) be the set of all non-negative zeros of \(AJ_m(x)+BxJ'_m(x)\) and let us denote \(\rho_{m,n}=z_{m,n}/a\) with \(a>0\). Then, the normalized radial functions \(\mathcal{J}^a_{m,n}\) on \([0,a]\) are defined by \[ \mathcal{J}^a_{m,n}=J_m(\rho_{m,n}r)/\sqrt{N^{(m)}_n(a)}, \] where \[ N^{(m)}_n(a)=\frac{a^2}{2} \left\{J'_m(z_{m,n})^2+\left(1-\frac{m^2}{z_{m,n}^2}\right)J_m(z_{m,n})^2 \right\}. \] Thus \(\{\mathcal{J}^a_{m,n}\}_{n\in\mathbb{N}}\) constitutes an orthonormal basis for the space \(L^2([0,a], r\,dr)\). Also, the normalized polar harmonics on \(\mathbb{B}_a=\{x\in\mathbb{R}^d: \|x\|_2\leq a\}\) are defined by \(\Psi^a_{m,n}(ru_\theta)=\mathcal{J}^a_{m,n}(r)\mathcal{Y}_m(\theta)\) and every function defined on \(\mathbb{B}_a\) can be expanded with respect to the family \(\{\Psi^a_{m,n}\}_{m,n}\).
Now, for every \(\xi\in L^2(\mathbb{B}_a)\) and \(k=(k_1,k_2)^T\in\mathbb{Z}^2\) let us consider \[ \hat{\xi}(k)=\int_0^a\int_0^{2\pi}\xi(r,\theta)e^{-i\pi a^{-1}(k_1\cos(\theta)+k_2\sin(\theta))}rdr d\theta \] and the subspace \(A_a=\{\xi\in L^2(\mathbb{B}_a)\, / \sum_{k\in\mathbb{Z}^2}|\hat{\xi}(k)|^2<\infty\}\).
In this context, the authors obtain some results among which we can highlight the following as a summary
{Theorem 1.} Let \(a>0\) and \(\xi\in A_a\) be a function. Suppose \(m\in\mathbb{Z}\) and \(n\in\mathbb{N}\). Then, \(C_{m,n}^a(\xi)=\sum_{k\in\mathbb{Z}^2}c_a(k;m,n)\hat{\xi}(k),\) with \[ c_a(k;m,n)=\sqrt{\pi}(-1)^ni^m\rho_{m,n}\frac{J_m(\pi|k|)e^{-im\Phi(k)}}{2\pi(\pi^2|k|^2-z^2_{m,n})}, \ \ \ \forall k=(k_1,k_2)^T\in\mathbb{Z}^2. \] {Theorem 2.} Let \(a>0\), \(f:\mathbb{R}^2\to\mathbb{C}\) be a function. Suppose \(R_a(f)\) is the restriction of \(f\) onto \(\mathbb{B}_a\) and \(R_a(f)\in A_a\). Then, \(R_a(f)=\sum_{m=-\infty}^\infty\sum_{n=1}^\infty c^a_{m,n}(f)\Psi^a_{m,n}\), where the series converges in \(L^2(\mathbb{B}_a)\) and \(c^a_{m,n}(f)=\sum_{k\in\mathbb{Z}^2}c_a(k;m,n)\hat{f}(a^{-1}k,a),\) \(\forall m\in\mathbb{Z}\), \(n\in\mathbb{N}\); also, \[ \hat{f}(w,a)=\int_0^a\int_0^{2\pi}f(r,\theta)e^{-i\pi r(w_1\cos(\theta)+w_2\\ \sin(\theta))}rdrd\theta, \\ \ \forall w=(w_1,w_2)^T\in\mathbb{R}^2. \] {Theorem 3.} Suppose \(a>0\) and \(b=a/2\). Let \(f_j:\mathbb{R}^2\to\mathbb{C}\) with \(j\in\{1,2\}\) be continuous functions supported on \(\mathbb{B}_b\). Assume \(m\in\mathbb{Z}\) and \(n\in\mathbb{N}\). Then, \[ c^a_{m,n}[f_1*f_2]=\sum_{k\in\mathbb{Z}^2}c_a(k;m,n)\hat{f}_1(a^{-1}k)\hat{f}_2(a^{-1}k), \] where \[ \hat{f}_j(w)=\int_0^a\int_0^{2\pi}f_j(r,\theta)e^{-i\pi r(w_1\cos(\theta)+w_2\\ \sin(\theta))}rdrd\theta, \\ \ \forall w=(w_1,w_2)^T\in\mathbb{R}^2. \] Therefore, the authors introduce a constructive method using Bessel-Fourier expansions for functions supported on disks, and the closure properties of such expansions under convolution are derived.

MSC:

43A50 Convergence of Fourier series and of inverse transforms
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A85 Harmonic analysis on homogeneous spaces
43A10 Measure algebras on groups, semigroups, etc.
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A20 \(L^1\)-algebras on groups, semigroups, etc.
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42A85 Convolution, factorization for one variable harmonic analysis
34B24 Sturm-Liouville theory
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References:

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