In order to understand the problem, I will first elaborate on the notation:
The definition of the functional ANOVA decomposition is given as follows. Let $F(x):[0,1]^d \rightarrow \mathbb{R}$ be $\mathbb{L}^2$ over $x=(x_1, \ldots, x_d)$. We can write $F(x) = f_0 + \sum_{i=1}^d f_i(x_i) + \sum_{i\neq j}f_{ij}(x_i,x_j)+\cdots$ In order to make the notation for the summation tractable, I will introduce the following multi-index notation: let $u \subseteq{1,\ldots,k}$, we denote by $x_u$, the subset of variables whose indexes are in $u$. Similarly, $x_{-u}$ indicates variables with indexes not in $u$. We can now write $F(x)$ as $F(x) = \sum_{u\subseteq\{1,\ldots,d\}} f_u(x_u)$ with $f_u$ depending only on $x_u$. For the sake of notational neatness, we write $u \subseteq d$ in place of $u \subseteq \{1,\ldots,d\}$.
Problems
- Why is $f_{1,\ldots,d}(x)$ orthogonal to the closed set of functions which are additive in $u$?
- How can lemma 1 be proved?