Consider $\Z_2$ as the set $\{\pm1\}$ with multiplication. For every $d\in\N$ and every subset $A$ of $\{1,\ldots,d\}$ put $w_A:\Z_2^d\rar\{\pm1\}$, $$ w_A(\o)\colon=\prod_{j\in A}\o_j,\quad w_\emptyset\colon=1~. $$ Then $w_A$, $A\sbe\{1,\ldots,d\}$, is an orthonormal basis for $L_2(\Z_2^d)$ -the measure being the normalized counting measure. These functions $w_A$ are called the Walsh functions. 2. If $P$ denotes the Markov operator of the simple random walk on $\Z_2^d$, then $$ Pw_A=\l_Aw_A \quad\mbox{and}\quad \l_A=\frac{|A^c|-|A|}{d}~. $$ In particular the multipliticity of $\l_n\colon=1-2n/d$ is ${d\choose n}$.
2. For $d=1$ we have $w_0\colon=w_\emptyset=1$, $w_1(\o)\colon=w_{\{1\}}(\o)=\o$ and thus $P_1w_0=w_0$, $P_1w_1=-w_1$. Now for $A\sbe\{1,\ldots,d\}$ we have $w_A=w_{r_1}\otimes w_{r_2}\otimes\cdots\otimes w_{r_d}$, where $r_j=1$ if $j\in A$ and otherwise $r_j=0$. Since $$ dP =P_1\otimes1\otimes\cdots\otimes1 +1\otimes P_1\otimes1\otimes\cdots\otimes1+\cdots +1\otimes\cdots\otimes1\otimes P_1 $$ we get $$ dPw_A =P_1w_{r_1}\otimes w_{r_2}\otimes\cdots\otimes w_{r_d} +\cdots +w_{r_1}\otimes\cdots\otimes P_1w_{r_d} =|\{j:r_j=0\}|w_A-|\{j:r_j=1\}|w_A =(|A^c|-|A|)w_A~. $$