Projection into Spaces Generated by Haar and Daubechies Scaling Functions

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In this Demonstration, selects which wavelet scaling function is used to project a function
into a collection of subspaces
of
. The approximation space
is defined as the span of
. Choices for
are the Haar scaling function and the Daubechies scaling function with two vanishing moments.
Contributed by: Sijia Liang and Bruce Atwood (February 2013)
(Beloit College)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Both scaling functions generate orthonormal bases. Thus the projection of into a
space is given by
, where
.
For students: The function is nonzero on
and the Haar scaling function is nonzero on
. From this, determine what values of
(as a function of
) make
nonzero. What about for the Daubechies scaling function that is nonzero on
? You can check your answers by looking at the source code.
Scaling functions are the basic building blocks for multiresolution analysis in wavelet theory. For more information see [1]. This Demonstration is based on an example from that book.
Reference
[1] D. K. Ruch and P. J. Van Fleet, Wavelet Theory: An Elementary Approach with Applications, Hoboken, NJ: John Wiley & Sons, 2009.
Permanent Citation