Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces?

Examples for what I have in mind: 

Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and $\emptyset \ne A\subset \{1,\ldots, n\}$.
Define $W_A:\{-1,1\}^n\to \{-1,1\}$ by 
$ W_A(x)= \prod_{i\in A} x_i.$ and $A_+,A_-\subset \{-1,1\}^n$ by 
$$A_+ =\{x\in\{-1,1\}^n:\ W_A(x)=1\}, \qquad A_- =\{x\in\{-1,1\}^n:\ W_A(x)=-1\}.$$
One can then define 
$$\|\hat f(A)\|^2= \frac{1}{4} d_X(b(f(A_+)), b(f(A_-)))^2 ,$$
where $b(U)$ is the barycenter of a finite subset $U\subset X$.
Is it possible to relate 
$$2^{-n} \sum_{x\in\{-1,1\}^n} d_X(f(x),b(f))^2$$
and
$$\sum_{\emptyset \ne A\subset \{1,\ldots, n\}} \|\hat f(A)\|^2\ ? $$
I.e., Does the Parseval identity hold in some weak sense?

How about relating 
$$\sum_{\emptyset \ne A\subset \{1,\ldots, n\}} |A|^2 \cdot \|\hat f(A)\|^2\ , $$
to 
$$2^{-n} n^2 \sum_{x\in\{-1,1\}^n} d_X(f(x),b(\{f(x\cdot e_j:\; j\in\{1,\ldots, n\}\}))^2, $$
where $e_j$ has value $1$ except in the $j$-th coordinate where it has value $-1$.
(This is a supposed analogue of the Parseval identity for $\Delta f$).