Shannon's sampling theorem and shifted sampling sequences

Question

Suppose that $f$ is a bandlimited function such that the Fourier transform of $f$ has support in $[-a/2,a/2]$. According to Shannon's sampling theorem, $f$ is determied from its samples at $\frac{1}{a}\mathbb{Z}$ and can be expanded in terms of a cardinal series. Thus, if $f(\tfrac{z}{a}) = 0$ for every $z \in \mathbb{Z}$ then $f$ is zero on the whole real line. I'm wondering if this property also holds if we shift the set $\frac{1}{a}\mathbb{Z}$ by a real number, i.e. if we consider the set $A=x+\frac{1}{a}\mathbb{Z}$ for some $x \in \mathbb R$. Does the implication $f(x+\tfrac{z}{a}) = 0$ for every $z \in \mathbb{Z} \implies f(t) = 0$ for every $t \in \mathbb{R}$ holds?

Answer 1

Yes; one can just set g(t) = f(x+t).