Laser Doppler anemometry (LDA) with burst spectrum analysis (BSA) is used to study the acoustic streaming generated in a cylindrical standing-wave resonator filled with air. The air column is driven sinusoidally at a frequency of approximately 310 Hz and the resultant acoustic-velocity amplitudes are less than 1.3 m/s at the velocity antinodes. The axial component of fluid velocity is measured along the resonator axis, across the diameter, and as a function of acoustic amplitude. The velocity signals are postprocessed using the Fourier averaging method [Sonnenberger et al., Exp. Fluids 28, 217-224 (2000)]. Equations are derived for determining the uncertainties in the resultant Fourier coefficients. The time-averaged velocity-signal components are seen to be contaminated by significant errors due to the LDA/BSA system. In order to avoid these errors, the Lagrangian streaming velocities are determined using the time-harmonic signal components and the arrival times of the velocity samples. The observed Lagrangian streaming velocities are consistent with Rott's theory [N. Rott, Z. Angew. Math. Phys. 25, 417-421 (1974)], indicating that the dependence of viscosity on temperature is important. The onset of streaming is observed to occur within approximately 5 s after switching on the acoustic field.