ABSTRACT We introduce DPSS Approximate lazY filtEriNg of foregroUnds (dayenu), a linear, spectral filter for H i intensity mapping that achieves the desirable foreground mitigation and error minimization properties of inverse co-variance weighting with minimal modelling of the underlying data. Beyond 21-cm power-spectrum estimation, our filter is suitable for any analysis where high dynamic-range removal of spectrally smooth foregrounds in irregularly (or regularly) sampled data is required, something required by many other intensity mapping techniques. Our filtering matrix is diagonalized by Discrete Prolate Spheroidal Sequences which are an optimal basis to model band-limited foregrounds in 21-cm intensity mapping experiments in the sense that they maximally concentrate power within a finite region of Fourier space. We show that dayenu enables the access of large-scale line-of-sight modes that are inaccessible to tapered discrete Fourier transform estimators. Since these modes have the largest SNRs,dayenu significantly increases the sensitivity of 21-cm analyses over tapered Fourier transforms. Slight modifications allow us to use dayenu as a linear replacement for iterative delay clean ing (dayenurest). We refer readers to the Code section at the end of this paper for links to examples and code.