Bessel operators of infinite order (Q2768773)

Let us consider a function \(\omega:[0,+\infty)\to[0,+\infty)\) which is continuous, increasing, \(\omega(0)=0, \omega(1)>1\), \(\lim_{x\to+\infty}\omega(x)=+\infty\). For \(x\geq 0\) we define \(\Omega(x)=\int_{0}^{x}\omega(\xi) d\xi\) and let \(\Omega(-x)=\Omega(x)\). Let functions \(\mu(x)\) and \(M(x)\) be defined in the same manner as \(\omega(x)\) and \(\Omega(x)\). The space \(W_{M}^{\Omega}\) is defined as a set of entire functions \(\varphi:\mathbb C\to\mathbb C\) such that \(\exists C>0, \exists a>0, \exists b>0, \forall z=x+iy\in\mathbb C: |\varphi(z)|\leq C\exp\{-M(ax)+\Omega(by)\}\). The space \(\overset\circ{W}_{M}^{\Omega}\) is defined as the set of all even entire functions from \(W_{M}^{\Omega}\). Let \(f(z)=\sum_{n=0}^{\infty}C_{2n}z^{2n}\) be an entire even function. We say that in the space \(\overset\circ{W}_{M}^{\Omega}\) the Bessel operator of the infinite order \(f(B)=\sum_{n=0}^{\infty}C_{2n}(-B)^{n}\) is defined if for arbitrary \(\varphi\in \overset\circ{W}_{M}^{\Omega}\) the series \((f(B)\varphi)(z)=\sum_{n=0}^{\infty}C_{2n}(-1)^{n}(B^{n}\varphi)(z)\) represents a function from \(\overset\circ{W}_{M}^{\Omega}\). Here \(B={d^2\over dz^2}+{2\nu+1\over z}{d\over dz},\;\nu>-1/2\). The authors prove that the operator \(f(B)\) is definite and continuous in \(\overset\circ{W}_{M}^{\Omega}\).