Extensions of the Heisenberg-Weyl inequality

On the Heisenberg-weyl Inequality

In 1927, W. Heisenberg demonstrated the impossibility of specifying simultaneously the position and the momentum of an electron within an atom.The well-known second moment Heisenberg-Weyl inequality states: Assume that f : R → C is a complex valued function of a random real variable x such that f ∈ L(R). Then the product of the second moment of the random real x for |f | and the second moment o...

On the Heisenberg-pauli-weyl Inequality

In 1927, W. Heisenberg demonstrated the impossibility of specifying simultaneously the position and the momentum of an electron within an atom.The following result named, Heisenberg inequality, is not actually due to Heisenberg. In 1928, according to H. Weyl this result is due to W. Pauli.The said inequality states, as follows: Assume thatf : R → C is a complex valued function of a random real ...

On the Sharpened Heisenberg-weyl Inequality

The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis states: Assume that f : R → C is a complex valued function of a random real variable x such that f ∈ L(R). Then the product of the second moment of the random real x for |f | and the second moment of the random real ξ for ∣∣∣f̂ ∣∣∣2 is at least E|f |2 / 4π, where f̂ is the Fourier transform...

On the Refined Heisenberg-weyl Type Inequality

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 ∈ L(R), where R = (−∞,∞). Then the product of the second moment of the random real x for |f | and the second moment of the random real ξ for ∣∣∣f̂ ∣∣∣2 is at least ER,|f |2 / 4π, where f̂ is the Fourier transform of f ...

A Necessary Condition for Weyl-Heisenberg Frames