How do you receive forgiveness from a friend with Asperger's Syndrome?

Same as from anyone else. However, a lot of us find that rational arguments and appeals to logic are more effective than apologies on an emotive base. So I'd suggest just saying "Look, here's what I did wrong, it was unintentional, sorry about that."

Also, don't confuse forgiveness with forgetting. People with Asperger's often have Excellent memories. Simply because they might not forget what happened does not mean that they have not forgiven you.

ADDENDUM:

In the year since I answered this question, I have got married. I've found that the last sentence above can be problematic; for me, missteps in the past are ancient history, while my wife can find them difficult. I often find myself mentioning things she feels embarrassed about, which she sees as a criticism; for me, however, they're simply relevant to the conversation, or random factoids that catch my attention at the time.

And that's what I suspect can flummox NTs. For Aspies - in my experience, of course - forgiven is forgiven. It's history; in the past, forgiven but remembered, and simply fact without emotional weight. For NTs, they can still be embarrassing or painful memories.