Fractional Legendre transformation

A new transformation is defined that connects a function and its Legendre transform by means of a continuous free parameter. The cyclic behaviour of consecutive Legendre transformations is reflected in the periodic dependence of the new transform on this parameter. This transformation opens new options wherever the conventional Legendre transformation is used (including mechanics, thermodynamics and optics) and is suggestively derived here by considering the geometrical-optics limit of a diffraction integral. The connection to a classical limit of the fractional Fourier transformation is also established and the mathematical and geometrical properties of the transformation are demonstrated.