This paper proposes a novel approach for line spectral estimation which combines Georgiou's filter bank (G-filter) with atomic norm minimization (ANM). A key ingredient is a Carathéodory--Fejér-type decomposition for the covariance matrix of the filter output. The resulting optimization problem can be characterized via semidefinite programming and contains the standard ANM for line spectral estimation as a special case. Simulations show that our approach outperforms the standard ANM in terms of recovering the number of spectral lines when the signal-to-noise ratio is no lower than 0 dB and the G-filter is suitably designed.