= Periodic phase change of a scaling function
If a periodic unimodular <function> $a$ and its inverse have absolutely summable <Fourier coefficients>, replacing $\widehat\varphi_0$ by $\widehat\varphi=a\widehat\varphi_0$ preserves the <orthonormal> integer-translate span and integrability of an integrable <scaling function>. In physical space this is an absolutely summable combination of <integer> <function translations>. The new <MRA low-pass filter> is $m(t)=a(2t)m_0(t)/a(t)$. A rough phase can therefore preserve the <multiresolution analysis> while changing <MRA low-pass filter> regularity.
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