Perturbations of the Haar wavelet

Abstract:

Let $m \in Z^+$ be given. For any $\varepsilon > 0$ we construct a function $f^{\{\varepsilon \}}$ having the following properties: (a) $f^{\{\varepsilon \}}$ has support in $[-\varepsilon , 1 + \varepsilon ]$. (b) $f^{\{\varepsilon \}} \in C^m(-\infty , \infty )$. (c) If $h$ denotes the Haar function and $0<\delta <\infty$, then $\Vert f^{\{\varepsilon \}} - h \Vert _{L^\delta (\mathcal R)} \le (1+2^\delta )^{1/\delta }(2\varepsilon )^{1/\delta }$. (d) $f^{\{\varepsilon \}}$ generates an affine Riesz basis whose frame bounds (which are given explicitly) converge to $1$ as $\varepsilon \rightarrow 0$.