Is this book----- you bought the other day?

首先,这是一个定语从句,the one 后面其实省略了一个that,that在定语从句中,作为从句中的bought的宾语,可以省略。再者,the one 是特指the other day 这一天借的书,不能省。最后,This is which you bought the other day,是不正确,应该是 This is what you bought the other day。因为which 不仅仅坐表语,还作为 bought 的宾语。