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Supports of Fourier transforms of scaling functions. (English) Zbl 1203.42008
Summary: We characterize the support \(G\) of the Fourier transform of the band-limited scaling function and give an approach to the construction of the scaling function. Based on the relations of translates of three point sets \(G\),\(\frac 12 G\) and \(G \setminus \frac 12 G\), we reveal an essential difference of scaling functions corresponding to MRA, weakly translation invariant MRA and translation invariant MRA. Moreover, we consider the case that support of the Fourier transform of the scaling function is a convex set. Finally, we show that the support of Fourier transform of any scaling function cannot be a closed ball.

MSC:
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42C15 General harmonic expansions, frames
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