Exponential B-splines and the partition of unity property
The authors consider the convolution of \(N\) exponential functions, denoted by \({\mathcal E}_N\), restricted to the interval \([0,1]\), i.e., \[ {\mathcal E}_N =e^{a_1 (\cdot)}\chi_{[0,1]}*e^{a_2 (\cdot)}\chi_{[0,1]}*\dotsb *e^{a_N (\cdot)}\chi_{[0,1]}, \] where \(a_i \in \mathbb R, i=1,2,\dotsc, N\). They provide an explicit formula for such a convolution, a compactly supported function. They show that the integer translates of \({\mathcal E}_N\) provide a partition of unity up to a multiplicative constant, i.e., \[ \sum_{k\in\,\mathbb Z} {\mathcal E}_N(x-k)= C,\quad x\in \mathbb R, \] for some constant \(C\) only if some of the \(a_i=0,\) \( i=1,2, \dotsc ,N,\) i.e., if \[ {\mathcal E}_N =e^{a_1 (\cdot)}\chi_{[0,1]}*e^{a_2 (\cdot)}\chi_{[0,1]}*\dotsb *e^{a_l (\cdot)}\chi_{[0,1]}*\underbrace{\chi_{[0,1]}*\chi_{[0,1]}*\chi_{[0,1]}}_{\text{\((N-l)\)-terms}}. \] The authors obtain an explicit formula for the constant \(C\) in case this happens. They show that if the modulation parameter \(b \leq \frac{1}{2N-1}\), then the system \(\{E_{mb}T_n{\mathcal E}_N\}_{m,n\in\,\mathbb Z}\) is a Gabor frame, where \((T_af)(x)=f(x-a)\) and \((E_bf)(x)=e^{2\pi i b x}f(x)\). In addition to that, if \(a_i=0\) for some \(i=1,2\dotsc, N\), then the above system has a dual frame \(\{E_{mb} T_n \tilde{\mathcal E}_N\}_{m,n \in \mathbb Z}\), where \(\tilde{\mathcal E}_N\) is some linear combination of integer translates of \({\mathcal E}_N\). An explicit construction for the coefficients in this linear combination is provided. They also prove that, if \(a_i\neq 0\) for all \(i=1,2, \dotsc ,N\) and \(a_i\neq -a_j\) for all \(i,j=1,2,\dotsc, N\), then no finite linear combination of \(\tilde{\mathcal E}_N\) can generate a dual Gabor frame.
scientific article; zbMATH DE number 6117987
scientific article