= Integral null-space ambiguity in expiratory measurements
\b[Integral null-space ambiguity in expiratory measurements] is the existence of different <volumetric flow rate> functions producing the same finite collection of cumulative expired <volumes>. Let $Q$ belong to an appropriate <function space> and let the measurement map be
$$
LQ=\left(\int_0^{T_1}Q(t)\,dt,\ldots,\int_0^{T_m}Q(t)\,dt\right).
$$
This is a <linear map>. Every perturbation $h$ in its <null space> leaves all the measurements unchanged: $L(Q+h)=LQ$.
The null space is large even when observations occur at several times. Put $T_0=0$. On any interval $[T_{j-1},T_j]$, choose a sine wave completing an integer number of cycles and set it to zero outside that interval. Its integral across that interval is zero, so every recorded cumulative <volume> is unchanged. Different cycle counts give infinitely many such functions. Sufficiently small amplitudes preserve positive <volumetric flow rate> wherever the baseline stays bounded away from zero.
An original example records <volumes> at 0.5, 1 and 2 s from a baseline <volumetric flow rate> of 1 L/s. Add $h(t)=0.4\sin(4\pi t)$ L/s only over the first 0.5 s. All three recorded <volumes> remain 0.5, 1 and 2 L, but the instantaneous <volumetric flow rate> ranges from 0.6 to 1.4 L/s during that interval. The added shape is invisible to these cumulative measurements.
A finite-dimensional model changes what can be identified. If $Q(t)=\sum_{j=1}^k a_j\phi_j(t)$ is restricted to known basis functions, the observations become $y=Ma$, where $M_{ij}=\int_0^{T_i}\phi_j(t)\,dt$. Unique coefficients require <matrix rank> $k$. For the stipulated linear-flow model $Q=a+bt$, two observations have rows $(T_i,T_i^2/2)$ and determinant
$$
\det M=\frac12T_1T_2(T_2-T_1).
$$
Distinct positive times therefore identify $a,b$ in that model. <Volumes> 1.5 L at 1 s and 4 L at 2 s yield $a=1$ L/s and $b=1\,\mathrm{L\,s^{-2}}$.
That model-based uniqueness is not recovery of an arbitrary waveform. The primary spirometry statement's complementary volume-time and flow-volume displays preserve information beyond a few timed <volumes>.<Standardization of Spirometry 2019 Update by Brian L. Graham and others>[Equipment, Display]{ref} State the allowed model class before treating a small set of summaries as a complete measurement.
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