An analogue of the Fourier transform will be introduced for all square integrable continuous martingale processes whose quadratic variation is deterministic. Using this transform we will formulate and prove a stochastic Heisenberg inequality.
【The Heisenberg inequality associated with the uncertainty principle is extended to an infinite dimensional abstract Wiener space (H, B) with an abstract Wiener measure p 1 . For ϕ∈ L 2 (p 1 ) and T ∈ L(B, H), it is shown that (※Equations, See Full-text), where F(sub) ϕ is the Fourier-Wiener transform of ϕ . The conditions when the equality holds also discussed.】
Some properties of the q-Fourier-sine transform are studied and q-analogues of the Heisenberg uncertainty principle is derived for the q-Fourier-cosine transform studied in \cite{FB} and for the q-Fourier-sine transform.
ABSTRACT. The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that f: R → C is a complex valued function of a random real variable x such that f ∈ L2 (R), where R = (−∞, ∞). Then the product of the second moment of the random real x for |f | 2 and the second moment of the random real ξ for ∣ f ∣ 2 is at least E 2 R,|f | 4π, where f is the Fourier transform of f, f (ξ) = ∫ R e−2iπξxf (x) dx and f (x) = ∫ R e2iπξx ˆ f (ξ) dξ, and E 2 R,|f | = ∫ R |f (x)|2 dx. This uncertainty relation is wellknown in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to the higher order moments for L2 (R) functions f. In this paper, a refined form of the generalized Heisenberg-Weyl type inequality is established.