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Resolution Spaces as Hierarchical

Third, take note of the hierarchical subspace structure of the resolution spaces . The Fourier transform of the basis elements, Eq.(2.90), for have compact support confined to . As was shown in Part (c) of Ex. 1.5.3 on page , these basis elements form a complete set. This means that if and only if its Fourier transform has support confined to . Next consider the vector space . The Fourier transform of its basis elements have support confined to

In fact, every element of enjoys this property. This implies that such elements also belong to . Thus one has

In other words, is a subspace of :

More explicitly, this inclusion property says that

Such a hierarchy of increasing subspaces is called a multiscale analysis of the space of square-integrable functions. A multiscale analysis is always derived from (i.e. based on) a scaling function . In our illustrative example this scaling function (father function'') is Shannon's sinc function, Eq.(2.88).

Next: Resolution Analysis as a Up: Multiresolution Analysis as Hierarchical Previous: Translation Followed by Compression   Contents   Index
Ulrich Gerlach 2010-12-09