Due to the combinatorial and modular nature of eukaryotic gene promoters, it is necessary to take both general and gene-class specific appoaches to the promoter recognition problem. For the general approach, we propose to divide the problem into two related discrimination sub-problems: (1) given a large genomic DNA sequence of 100 kb or larger, how to localize a promoter into a 1-2 kb (extended-promoter) region; (2) given such an extended-promoter region, how to further localize a Transcriptional Start Site into a 50-100 bp (Core-promoter) region? We shall describe some recent results on the second sub-problem where we have applied the Quaratic Discriminant Analysis using positional k-tuple measures.