  The fragility of quantum systems makes them ideally suited for sensing
applications at the nanoscale. However, interpreting the output signal of a
qubit-based sensor is generally complicated by background clutter due to
out-of-band spectral leakage, as well as ambiguity in signal origin when the
sensor is operated with noisy hardware. We present a sensing protocol based on
optimally band-limited "Slepian functions" that can overcome these challenges,
by providing narrowband sensing of ambient dephasing noise, coupling additively
to the sensor along the ${z}$-axis, while permitting isolation of the target
noise spectrum from other contributions coupling along a different axis. This
is achieved by introducing a finite-difference control modulation, which
linearizes the sensor's response and affords tunable band-limited "windowing"
in frequency. Building on these techniques, we experimentally demonstrate two
spectral estimation capabilities using a trapped-ion qubit sensor. We first
perform efficient experimental reconstruction of a "mixed" dephasing spectrum,
composed of a broadband $1/f$-type spectrum with discrete spurs. We then
demonstrate the simultaneous reconstruction of overlapping dephasing and
control noise spectra from a single set of measurements, in a setting where the
two noise sources contribute equally to the sensor's response. Our approach
provides a direct means to augment quantum-sensor performance in the presence
of both complex broadband noise environments and imperfect control signals, by
optimally complying with realistic time-bandwidth constraints.
