Define
(1)
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and
(2)
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for a nonnegative integer and .
So, for example, the first few values of are
(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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Then a function can be written as a series expansion by
(10)
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The functions and are all orthogonal in , with
(11)
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(12)
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for in the first case and in the second.
These functions can be used to define wavelets. Let a function be defined on intervals, with a power of 2. Then an arbitrary function can be considered as an -vector , and the coefficients in the expansion can be determined by solving the matrix equation
(13)
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for , where is the matrix of basis functions. For example, the fourth-order Haar function wavelet matrix is given by
(14)
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(15)
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