Translation (a)
Dilation (b)
The Mexican hat wavelet is given by: $$\psi(t) = \frac{2}{\sqrt{3\sigma}\pi^{1/4}} \left(1 - \left(\frac{t}{\sigma}\right)^2\right) e^{-\frac{t^2}{2\sigma^2}}$$ The continuous wavelet transform is given by: $$T(a,b) = w(a)\int_{-\infty}^{\infty} x(t) \psi^{*}\left(\frac{t-b}{a}\right)dt$$ The Fourier transform of the Mexican hat wavelet is given by: $$\hat{\psi}(w) = \frac{2 \sigma^{5/2}}{\sqrt{3}\pi^{1/4}} w^2 e^{-\frac{\sigma^2 w^2}{2}}$$ The continuous wavelet transform can be written using Fourier transform with: $$T(a,b) = \sqrt{a} \int_{-\infty}^{\infty} \hat{x}(w) \hat{\psi}^{*}(aw) e^{i(2\pi w)b}dw$$