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**A family of polynomial spline wavelet transforms.**
*(English)*
Zbl 0768.41012

Summary: This paper presents an extension of the family of orthogonal Battle/Lemarié spline wavelet transforms with emphasis on filter bank implementation. Spline wavelets that are not necessarily orthogonal within the same resolution level, are constructed by linear combination of polynomial spline wavelets of compact support, the natural counterpart of classical \(B\)-spline functions. Mallat’s fast wavelet transform algorithm is extended to deal with these non-orthogonal basis functions. The impulse and frequency resonances of the corresponding analysis and synthesis filters are derived explicitly for polynomial spline of any order \(n\) (\(n\) odd). The link with the general framework of biorthogonal wavelet transforms is also made explicit. The special cases of orthogonal, \(B\)-spline, cardinal and dual wavelet are considered in greater detail. The \(B\)-spline (respectively dual) representation is associated with simple FIR binomial synthesis (respectively analysis (respectively synthesis) filters. The cardinal representation provides a sampled representation of the underlying continuous functions (interpolation property). The distinction between cardinal and orthogonal representation vanishes as the order of the spline is increased; both wavelets tend asymptotically to the bandlimited sinc-wavelet. The distinctive features of these various representtions are discussed and illustrated with a texture analysis example.

### MSC:

41A15 | Spline approximation |

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |