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This is Part I of a two-part paper series that studies the use of the proportional fairness (PF) utility function as the basis for resource allocation and scheduling in multichannel multi-rate wireless networks. The contributions of Part I are threefold. (i) We present the fundamental properties and physical/economic interpretation of PF optimality. We show that PF leads to equal airtime allocation to users for the singlechannel case; and equal equivalent airtime allocation to users for the multi-channel case. In addition, we also establish the Pareto efficiency of joint-channel PF optimal solution (the formulation of interest to us in this paper), and its superiority over the individual-channel PF optimal solution in that the individual user throughputs of the former are all equal to or greater than the corresponding user throughputs of the latter. (ii) Second, we derive characteristics of joint-channel PF optimal solutions useful for the construction of PF-optimization algorithms. In particular, we show that a PF solution typically consists of many zero airtime assignments when the difference between the number of users U and the number of channels S, |U - S|, is large. We present several PF-optimization algorithms, including a fast algorithm that is amenable to parallel implementation. (iii) Third, we study the use of PF utility for resource allocation in large-scale WiFi networks consisting of many adjacent wireless LANs. We find that the PF solution simultaneously achieves higher system throughput, better fairness, and lower outage probability with respect to the default solution given by today's 802.11 commercial products. Part II of this paper series extends our investigation to the time-varying-channel case in which the data rates enjoyed by users over the channels vary dynamically over time.