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Given $X$ a Hilbert space, $\omega$ a modulus of continuity, $E$ an arbitrarysubset of $X$, and functions $f:E\to\mathbb{R}$, $G:E\to X$, we providenecessary and sufficient conditions for the jet $(f,G)$ to admit an extension$(F, \nabla F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1,\omega}(X)$, by means of a simple explicit formula. As a consequence of thisresult, if $\omega$ is linear, we show that a variant of this formula providesexplicit $C^{1,1}$ extensions of general (not necessarily convex) $1$-jetssatisfying the usual Whitney extension condition, with best possible Lipschitzconstants of the gradients of the extensions. Finally, if $X$ is asuperreflexive Banach space, we establish similar results for the classes$C^{1, \alpha}_{\textrm{conv}}(X)$.

Explicit formulas for $C^{1, 1}$ and $C^{1, \omega}_{\textrm{conv}}$ extensions of $1$-jets in Hilbert and superreflexive spaces