Extension to of a problem from Rudin to unbounded interval in $\mathbb{R}$

Question

The question that has been asked here for several times in various forms is somewhat like this (cf. the following links):

Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

Also,

Integral of a continuous function

Again,

Prove that $f(x) = 0$ on the interval $[0,1]$

And so on!!

Now just being curious I am asking for similar result (BUT on unbounded domain) if at all true, in the form of following problem:

Suppose $v \in C([0,\infty))$ and $v$ vanishes at infinity. If $v$ is assumed to be square integrable and $lim_{x \to \infty} \int_{0}^{x}(x-y)^{k} v(y) dy = 0 \quad \forall k \in \mathbb{N}\cup{0}$, can we prove or disprove that $v$ is identically equal to 0 ??

Any form of solution/discussion is much appreciated!!