A grammar book I have says (b) can be used instead of (a).
(a) He was ashamed of having asked such a silly question.
(b) He was ashamed of asking such a silly question.
If it is true, can I say like (c) even when the sentence has a phrase that shows past?
(c) He was ashamed of asking such a silly question the day before.
He was ashamed of asking such a silly question the day before. Yes, this is fine. You might say that the perfect would be redundant here because the past/anterior meaning is conveyed by the day before.
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He was ashamed of asking such a silly question the day before.
Yes, this is fine. You might say that the perfect would be redundant here because the past/anterior meaning is conveyed by the day before.