Stability condition and Riesz bounds for exponential splines (Q6587471)

Let \(p\in\mathbb{R}\setminus\{0\}\) and let \(m\in\mathbb{N}\). The author introduces a family of exponential splines \(U_{m,p}\) recursively as follows:\N\begin{align*}\NU_{1,p} (x) &:= \begin{cases} \frac{p e^{p x}}{e^p-1}, & x\in [0,1);\\\N0, & \text{else.} \end{cases}\\\NU_{m+1, p} (x)&:= (U_{m,p} * \chi_{[0,1)}) (x), \quad m\in \mathbb{N},\N\end{align*}\Nwhere \(*\) denotes convolution and \(\chi_{[0,1)}\) the indicator function of \([0,1)\). The last equation can be rewritten in the form\N\[\NU_{m+1, p} = Q_{m} * \varphi_p,\N\]\Nwhere \(Q_m\) denotes the cardinal B-splines of order \(m\) and \(\varphi_p := U_{1,p}\).\N\NThe stability of the integer translates of \(U_{m,p}\) for arbitrary \(m\) and \(p\) as defined above is established and the Riesz bounds computed. Furthermore, the method employed to obtain this result allows the computation of the Riesz bounds for the convolution of a cardinal B-splines \(Q_{m}\) of arbitrary order and a function with an appropriate Fourier transform.