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 $\Delta f$).
