Here, we present a novel algorithm for frequent itemset mining for streaming data (FIM-SD). For the past decade, various FIM-SD methods in one-pass approximation settings have been developed to approximate the frequency of each itemset. These approaches can be categorized into two approximation types: parameter-constrained (PC) mining and resource-constrained (RC) mining. PC methods control the maximum error that can be included in the frequency based on a pre-defined parameter. In contrast, RC methods limit the maximum memory consumption based on resource constraints. However, the existing PC methods can exponentially increase the memory consumption, while the existing RC methods can rapidly increase the maximum error. In this study, we address this problem by introducing the notion of a condensed representation, called a $Δ$-covered set, to the RC approximation. This notion is regarded as an extension of the closedness compression and when $Δ= 0$, the solution corresponds to an ordinary closed itemset. The algorithm searches for such approximate closed itemsets that can restore the frequent itemsets and their frequencies under resource constraint while the maximum error is bounded by an integer, $Δ$. We first propose a one-pass approximation algorithm to find the condensed solution. Then, we improve the basic algorithm by introducing a unified PC-RC approximation approach. Finally, we empirically demonstrate that the proposed algorithm significantly outperforms the state-of-the-art PC and RC methods for FIM-SD.