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

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 $$\psi(0)=0$$?

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()


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