Smooth Parseval frames for \(L^{2}(\mathbb{R})\) and generalizations to \(L^{2}(\mathbb{R}^d)\) (Q2867972)

A wavelet (or a Parseval frame) set \(E\) is a Lebesgue measurable subset in \(\mathbb{R}^d\) such that \(\{2^{jd/2}\psi_E(2^j \cdot-k) : j\in \mathbb{Z}, k\in \mathbb{Z}^d\}\) forms an orthonormal basis (or a Parseval frame) for \(L^2(\mathbb{R}^d)\), where \(\psi_E\) is the inverse Fourier transform of \(\chi_E\) -- the characteristic function of the set \(E\). However, \(\chi_E\) is a discontinuous function and therefore, \(\psi_E\) does not have rapid decay in the space domain. Wavelets and Parseval frames with rapid decay and smoothness are of great interest in wavelet analysis. Therefore, it is an important problem in wavelet analysis to find smooth functions \(g\) such that \(g\) can be arbitrarily close to \(\chi_E\) and \(\{2^{jd/2}\psi(2^j \cdot-k) : j\in \mathbb{Z}, k\in \mathbb{Z}^d\}\) forms a Parseval frame for \(L^2(\mathbb{R}^d)\), where \(\psi\) is the inverse Fourier transform of \(g\). In dimension one, this problem has been addressed in the reviewer's Master's thesis in 1994 by using a smoothing technique. However, wavelet sets in higher dimensions often have much more complicated geometric structure and this still remains as a challenging problem about how to approximate the characteristic function of a wavelet or Parseval frame set in high dimensions by smooth functions with rapid decay in the space domain. For a particular set \([-2a,2a]^2 \backslash [-a,a]^2\) with \(0<a<1/4\), the author made an interesting advance in this paper by showing that it is still possible to approximate the characteristic functions of such Parseval frame sets by smooth functions (see Proposition~2.6) using a modified smoothing technique as in the case of dimension one. A construction of smooth Parseval frames in higher dimensions is also addressed in Section 3. The discussion on the frame bound gaps in Section~4 also indicates the difficulty in smoothing the characteristic functions of Parseval frame sets by smooth Parseval frame generators \(g\) with rapid decay in the space domain.