Tensor transform of Boolean functions and related algebraic and probabilistic properties

We introduce a tensor transform for Boolean functions that covers the algebraic normal and Walsh transforms but which also allows for the definition of new, probabilistic and weight transforms, relating a function to its bias polynomial and to the weights of its subfunctions respectively. Our approach leads to easy proofs for some known results and to new properties of the aforecited transforms. Finally, we present a new probabilistic characteristic of a Boolean function that is defined by its algebraic normal and probabilistic transforms over the reals