A scheme for interpolation by Hankel translates of a basis function
The authors develop a variational approach following \textit{W. Light} and \textit{H. Wayne} [Numer. Math. 81, No. 3, 415--450 (1999; Zbl 0918.65007)] for interpolation by Hankel tranforms of a basis function. Let \(I=(0,\infty)\), \(w\in C(I)\), \(w(x)>0\) be a weight; let \(\mu\geq -1/2\), \(m\in\mathbb{N}_0\), \(\pi_{\mu,m-1}\) the linear space of Müntz polynomials spanned by the monomials \(p_{\mu,n}(t)=t^{2n+\mu+1/2}\) \((t\in I,n\in\mathbb{Z}_+,0\leq n\leq m-1)\). Let \(S_{\mu}=x^{-\mu-1/2}Dx^{2\mu+1}Dx^{-\mu-1/2}\) denote the Bessel differential operator, \(J_{\mu}\) the Bessel function of the first kind and order \(\mu\). \(\mathcal{H}_{\mu}'\) is the dual to the Fréchet space \(\mathcal{H}_{\mu}\) of all smooth complex valued functions \(\varphi\) such that \[ \nu_{\mu,p}(\varphi)=\max_{0\leq k\leq p}\sup_{x\in I}\left|(1+x^2)^p(x^{-1}D)^kx^{-\mu-1/2}\varphi(x)\right|<\infty,\quad p\in\mathbb{Z}_+, \] with the Hankel integral transform \[ (h_{\mu}\varphi)(x)=\int_0^{\infty}\varphi(t)\mathcal{J}_{\mu}(xt)\, dt,\quad \mathcal{T}_{\mu}(z)=z^{1/2}J_{\mu}(z), \] as an automorphism on \(\mathcal{H}_{\mu}\). \(h_{\mu}'\) is defined by transposition. Let further \[ L_{\mu,l}^1=\left\{v:\int_0^{a}|v(t)|t^{\mu+1/2}\,dt<\infty\, (a>0)\right\}; \] \[ L_{\mu,w}^2=\left\{u:\left(\int_0^{\infty}|u(t)|^2w(t)t^{\mu+1/2}\,dt\right)^{1/2}=\| u\|_{\mu,w}<\infty\right\}; \] \[ Y=Y_0=\{f\in\mathcal{H}'_{\mu}:\, h_{\mu}'f\in L_{\mu,l}^1\cap L_{\mu,w}^2\}\text{ with the norm } \| f\|_Y=\| h_{\mu}'f\|_{\mu,w}; \] \[ Y_m=\{f\in\mathcal{H}'_{\mu}: S_{\mu}^mf\in Y\} \text{ with the seminorm } |f|_m=\| S_{\mu}^mf\|_Y. \] If \(1/w\in L_{\mu,l}^1\) and there exists \(\gamma\in\mathbb{R}\) for which \(1/w(t)=\mathcal{O}(t^{-\gamma})\) as \(t\to\infty,\) and \(n\in \mathbb{Z}_+\) is such that \(4n<4m+\gamma-\mu -3/2\), then it is possible to define a basis distribution \(\phi\in\mathcal{H}_{\mu}'\) from the identity \(t^{4m}(h_{\mu}'\phi)(t)=1/w(t)\). The authors prove also that \(Y_m\) is embedded in a space of continuous functions, so that \(Y_m\) may be endowed with a Hilbert space structure where the point evaluations, i.e., \(R_{n,x}\in Y_m \) satisfying \((f,R_{n,x})=(S_{ \mu}^nf)(x)\) \((f\in Y_m),\) are continuous linear functionals. Moreover, they find explicitly the Fréchet-Riesz representatives for \(R_{n,x}\) with the help of the Hankel translates \[ (\tau_y\phi)(x)=\int_0^{\infty}\phi(z)D_{\mu}(x,y,z)\,dz, \] where \[ D_{\mu}(x,y,z)=\int_0^{\infty}t^{-\mu-1/2}\mathcal{J}_{\mu}(xt)\mathcal{J}_{\mu}(yt)\mathcal{J}_{\mu}(zt)\,dt \] is the so-called Delsarte kernel, and suitable Müntz polynomials from \(\pi_{\mu,m-1}.\) This result allows to obtain an error estimate for the related minimal norm interpolant to \(f\) on a set \(\{a_1,\ldots, a_l\}\subset I,\) i.e., the element \(Uf\in Y_m\) such that \((Uf)(a_j)=f(a_j)\) \((j=1,\dotsc, l)\), and, if \(g\in Y_m\) is another interpolant, then \(\| g\|_{Y_m}\geq \| Uf\|_{Y_m}\). Examples (including numerics) are given.
scientific article; zbMATH DE number 6112920
scientific article