It depends who you ask, Nyggus.
Some people say you must never end a sentence with a preposition. You should always say: "It is the only method on which the authors based their interpretation" and "This is the phenomenon about which we are to infer" (although I'm not sure if "infer" is the right word here).
Other people say that this "rule" was invented a few hundred years ago in order to make English grammar more like Latin grammar, and that this rule is wrong. They say that putting prepositions at the end of sentences is one of the things that makes English English, and we shouldn't try to use Latin grammar rules for English.
Winston Churchill took the latter view: it is said that he once told somebody, "This is the kind of English up with which I will not put." However, he was exaggerating; as tdol would say, the sentence would be better written, "This is the kind of English with which I will not put up," although that doesn't sound much better to me.
Personally, I share the view that it's OK to end a sentence with a preposition. However, because there are lots of people who insist on the rule, you should try not to end a sentence with a preposition, especially in formal writing, e.g. if you're applying for a job. Not everybody thinks it's OK to break this rule, but everyone agrees that "It is the only method on which..." and "This is the phenomenon about which..." is good English -- so be safe, and use this construction.