Beta Wavelet Spectrum
The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function .
Let denote the Fourier transform pair associated with the wavelet.
This spectrum is also denoted by for short. It can be proved by applying properties of the Fourier transform that
where .
Only symmetrical cases have zeroes in the spectrum. A few asymmetric beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold
Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by
This is henceforth referred to as an -order beta wavelet. They exist for order . After some algebraic handling, their closed-form expression can be found:
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