Is it possible that the value of a continuous mother wavelet at origin is zero, i.e. $\psi(t=0)=0$?

Question

According to Fourier transform, a continuous wavelet could be written as $$ \psi(t)=\frac{1}{2\pi}\int\hat\psi(k)\text{e}^{-ikt}\text{d}k $$

From the equation above, we know that $\psi(t=0)$ is $$ \psi(0)=\frac{1}{2\pi}\int\hat\psi(k)\text{d}k $$

Is it possible that $\psi(0)=0$?

Answer 1

Yes: a sufficient criterion for a valid wavelet is being zero mean in time domain, i.e. $\hat \psi (0) = 0$. For $\psi(0) = 0$ we require that $\hat \psi$ sums to zero; one example is higher order Generalized Morse Wavelets - from Olhede & Walden,

(Though above is only approximately zero-sum). Such wavelets can be admissible and analytic, enabling CWT inversion. Trivial Python example below; GMWs are also implemented in ssqueezepy.

Code

import numpy as np
import matplotlib.pyplot as plt
from numpy.fft import ifft, ifftshift

t = np.linspace(0, 1, 256, endpoint=False)
wf = np.exp(-(t - .08)**2 * 4096)
wf -= np.roll(wf, 25)    # shift by 25 samples and subtract
w = ifftshift(ifft(wf))  # take to time domain and center

plt.plot(wf)
plt.show()

plt.plot(w.real)
plt.plot(w.imag)
plt.show()

Answer 2

Yes, perfectly possible. That means that the mean of the function $\hat{\psi}(k)$ is zero.