The upper panel is for the Gaussian and the wavelet function; the lower panel is for their corresponding Fourier transforms. We see that the Gaussian is a low-pass filter, while the wavelet is a band-pass filter."
Figure 1.
Figure 2.
Upper panel: the 1D density contrast field of dark matter. Lower panel: the corresponding wavelet scalogram of the 1D density contrast field. The data is taken from the IllustrisTNG simulation."
Figure 2.
Figure 3.
The Fourier and wavelet power spectrum of the 1D density contrast field for dark matter. The left vertical axis is for the Fourier power spectrum, and the right axis is for the wavelet power spectrum. The data is taken from the IllustrisTNG simulation."
Figure 3.
Figure 4.
The Fourier and wavelet power spectrum of the 1D velocity field for baryonic fluid. The left vertical axis is for the Fourier power spectrum, and the right axis is for the wavelet power spectrum. The wavelet energy spectrum is also shown for comparison. The data is taken from the IllustrisTNG simulation."
Figure 4.
Figure 5.
The functional form of the counter-example (upper panel), and its Fourier transform (lower panel) in section 5, shown as blue lines. The Gaussian-derived wavelet and its Fourier transform are also shown for comparison (red lines)."
Figure 5.
Figure B1.
Three different smoothing functions and their corresponding wavelets. Upper left panel: Gaussian (red line), Meyer scaling (blue line) and Epanechnikov (green line) function in real space. Upper right panel: the wavelet functions derived from these smoothing functions. Lower left panel: Fourier transforms of the three smoothing functions. Lower right panel: Fourier transforms of the wavelets derived from the corresponding smoothing functions."
Figure B1.
Figure C1.
The relationship between our wavelet power spectrum and the traditional CWT power spectrum. Upper panel: the Gaussian-derived wavelet power spectrum Sd(w) (red line), and the traditional wavelet power spectrum ${S}_{{\rm{d}}}^{t}(a=1/w)$ (blue line). Lower panel: the ratio of ${S}_{{\rm{d}}}^{t}(w)$ to LbwSd(w)."
)
