  We consider estimation of a functional of the data distribution based on
i.i.d. observations. We assume the target function can be defined as the
minimizer of the expectation of a loss function over a class of $d$-variate
real valued cadlag functions that have finite sectional variation norm. For all
$k=0,1,\ldots$, we define a $k$-th order smoothness class of functions as
$d$-variate functions on the unit cube for which each of a sequentially defined
$k$-th order Radon-Nikodym derivative w.r.t. Lebesgue measure is cadlag and of
bounded variation. For a target function in this $k$-th order smoothness class
we provide a representation of the target function as an infinite linear
combination of tensor products of $\leq k$-th order spline basis functions
indexed by a knot-point, where the lower (than $k$) order spline basis
functions are used to represent the function at the $0$-edges. The $L_1$-norm
of the coefficients represents the sum of the variation norms across all the
$k$-th order derivatives, which is called the $k$-th order sectional variation
norm of the target function. This generalizes the zero order spline
representation of cadlag functions with bounded sectional variation norm to
higher order smoothness classes. We use this $k$-th order spline representation
of a function to define the $k$-th order spline sieve minimum loss estimator
(MLE), Highly Adaptive Lasso (HAL) MLE, and Relax HAL-MLE. For first and higher
order smoothness classes, in this article we analyze these three classes of
estimators and establish pointwise asymptotic normality and uniform convergence
at dimension free rate $n^{-k^*/(2k^*+1)}$ up till a power of $\log n$
depending on the dimension, where $k^*=k+1$, assuming appropriate
undersmoothing is used in selecting the $L_1$-norm. We also establish
asymptotic linearity of plug-in estimators of pathwise differentiable features
of the target function.
