To check that an object is of this class, use functions
is.Lfd or int2Lfd. Linear differential operator objects are often used to
define roughness penalties for smoothing towards a
"hypersmooth" function that is annihilated by the operator.
For example, the harmonic acceleration operator used in the
analysis of the Canadian daily weather data annihilates linear
combinations of $1, sin(2 pi t/365)$ and $cos(2 pi t/365)$,
and the larger the smoothing parameter, the closer the smooth
function will be to a function of this shape.
Function pda.fd estimates a linear differential
operator object that comes as close as possible to annihilating
a functional data object.
A linear differential operator of order $m$ is a
linear combination of the derivatives of a functional
data object up to order $m$. The derivatives of
orders 0, 1, ..., $m-1$ can each be multiplied
by a weight function $b(t)$ that may or may not vary with
argument $t$.
If the notation $D^j$ is taken to
mean "take the derivative of order $j$", then a linear
differental operator $L$ applied to function $x$
has the expression
$Lx(t) = b_0(t) x(t) + b_1(t)Dx(t) + ... + b_{m-1}(t) D^{m-1} x(t)
+ D^mx(t)$