Geodesic
On the inheritance of properties of Boolean functions under restrictions
Diskretnaya Matematika, Tome 14 (2002) no. 2, pp. 9-19.

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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. 
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     title = {On the inheritance of properties of {Boolean} functions under restrictions},
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O. A. Logachev; A. A. Sal'nikov; V. V. Yashchenko. On the inheritance of properties of Boolean functions under restrictions. Diskretnaya Matematika, Tome 14 (2002) no. 2, pp. 9-19. http://geodesic.mathdoc.fr/item/DM_2002_14_2_a1/

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