Duality (mathematics) - Analytic Dualities

Analytic Dualities

In analysis, problems are frequently solved by passing to the dual description of functions and operators.

Fourier transform switches between functions on a vector space and its dual:

and conversely

If f is an L2-function on R or RN, say, then so is and . Moreover, the transform interchanges operations of multiplication and convolution on the corresponding function spaces. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groups R (or RN etc.): any character of R is given by ξ↦ e−2πixξ. The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions of quantum mechanical systems in terms of coordinate and momentum representations.

• Laplace transform is similar to Fourier transform and interchanges operators of multiplication by polynomials with constant coefficient linear differential operators.
• Legendre transformation is an important analytic duality which switches between velocities in Lagrangian mechanics and momenta in Hamiltonian mechanics.