On the inheritance of properties of Boolean functions under restrictions

For a property $\mathcal P$ of Boolean functions, a Boolean function $f(x)$, $x\in V_n$, and a subspace $H$ of the space $V_n$ of all $n$-tuples of zeros and ones, we consider the set of all restrictions of the Boolean function $f(x)$ onto the cosets of $V_n$ with respect to $H$. If the function $f(x)$ itself and all its $2^{n-\dim H}$ restrictions possess the property $\mathcal P$, we say that the property $\mathcal P$ is inherited under the restrictions of the Boolean function $f(x)$ and consider it as a new derived property.
In this paper, this approach is applied to the following property of Boolean functions: the value $\hat f(\alpha)/2^n$, where $\hat f(\alpha)$ is the Walsh–Hadamard coefficient, is fixed; the corresponding derived property is called the $(H,\alpha)$-stability.
We give convenient criteria for $(H,\alpha)$-stability in terms of zeros of the Walsh–Hadamard coefficients, and establish relations between the $(H,\alpha)$-stability, correlation immunity, and $m$-resiliency.
This research was supported by the Russian Foundation for Basic Research, grants 99–01–00929 and 99–01–00941.