  We deal with the problem of optimal estimation of the linear functionals
constructed from the missed values of a continuous time stochastic process
$\xi(t)$ with periodically stationary increments at points $t\in[0;(N+1)T]$
based on observations of this process with periodically stationary noise. To
solve the problem, a sequence of stochastic functions $
\{\xi^{(d)}_j(u)=\xi^{(d)}_j(u+jT,\tau),\,\, u\in [0,T), j\in\mathbb Z\}. $ is
constructed. It forms a $L_2([0,T);H)$-valued stationary increment sequence
$\{\xi^{(d)}_j,j\in\mathbb Z\}$ or corresponding to it an (infinite
dimensional) vector stationary increment sequence
$\{\vec\xi^{(d)}_j=(\xi^{(d)}_{kj}, k=1,2,\dots)^{\top}, j\in\mathbb Z\}$. In
the case of a known spectral density, we obtain formulas for calculating values
of the mean square errors and the spectral characteristics of the optimal
estimates of the functionals. Formulas determining the least favorable spectral
densities and the minimax (robust) spectral characteristics of the optimal
linear estimates of functionals are derived in the case where the sets of
admissible spectral densities are given.
