Are there distributions on the unit circle, that cannot be described as a sum of smooth functions(and their primitives), dirac deltas and their derivatives, and a linear sum of such?

Here I am considering the unit circle instead of the real line to avoid infinite sums like the comb, ie, $\sum a_n \delta(x - n)$.

The responses to another question clarifies that the best known examples of distributions that are not measures, are the derivatives of the delta and such. What I want to know is: Is that the only way a distribution is not a measure?

Are there distributions that are not measures, their derivatives, and their derivatives, and so on?

Are there distributions on the unit circle, that cannot be described as a sum of smooth functions(and their primitives), dirac deltas and their derivatives, and a linear sum of such?

Here I am considering the unit circle instead of the real line to avoid infinite sums like the comb, ie, $\sum a_n \delta(x - n)$.

The responses to another question clarifies that the best known examples of distributions that are not measures, are the derivatives of the delta and such. What I want to know is: Is that the only way a distribution is not a measure?