Weapon normalization?

Hi folks - maybe my google & forum fu is failing, but I can't seem to find an answer to the following question, and figured y'all would know:

Are 'on hit'-style effects normalized for weapon speed? For example, suppose I have a 3 damage, .33 swings per second weapon and a 1 damage, 1 swing per second weapon (i.e. both are 1 dps)

Would a +1 damage gem make the first weapon 4dps and the 2nd weapon 6 dps?

Similarly, would a +1 healing on hit, assuming 100% hit rate, make the first weapon heal 1hps and the 2nd weapon heal 3 hps?
the answer is yes.
faster weapons also generate spirit faster (due to hitting more times).

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