Contact Geometry and Linear Differential Equations by Vladimir E. Nazaikinskii, Victor E. Shatalov, Boris Yu.

By Vladimir E. Nazaikinskii, Victor E. Shatalov, Boris Yu. Sternin

The target of the sequence is to provide new and demanding advancements in natural and utilized arithmetic. good verified locally over 20 years, it bargains a wide library of arithmetic together with numerous vital classics.

The volumes offer thorough and particular expositions of the equipment and concepts necessary to the themes in query. furthermore, they impart their relationships to different elements of arithmetic. The sequence is addressed to complicated readers wishing to entirely research the topic.

Editorial Board

Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil
Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia
Walter D. Neumann, Columbia collage, big apple, USA
Markus J. Pflaum, collage of Colorado, Boulder, USA
Dierk Schleicher, Jacobs collage, Bremen, Germany

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Contact Geometry and Linear Differential Equations

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Extra info for Contact Geometry and Linear Differential Equations

Example text

Similarly. of n is even, we obtain — f(x)=i1 — — I)! /2 2i"(n — 1)! p. f(p)w(p) (x . p)fl f 3 The latter integral can be rewritten in the more usual form as an integral over the unit sphere in the space The commutation formulas for the Radon transform can also be obtained from the corresponding formulas for the Fourier transform of homogeneous functions. To do so, we note that = 44 1. Homogeneous functions, Fourier transformation, and contact structures and hence, af(x') (j*)_I { j = f(x') E F.

Fourier transformation of homogeneous functions The Fourier transformation of homogeneous functions has been studied thoroughly, beginning with the fundamental paper of Gel' fand and Shapiro ([GSha 1J). However. we present here a result which does not seem to be covered by the existing investigations. We show that several very natural properties determine this transformation up to a scalar factor. Namely, for functions of fixed degree and parity, it suffices to require continuity of the transformation and its natural behaviour under linear variable changes.

Homogeneous functions. Fourier transformation, and contact structures The existence of a nonzero function x(s) which is homogeneous of degree 1 gives us an opportunity to identify the contactization C of the symplectic space S with the submanifold in S which is determined by the equation = I. In this case, any Hamiltonian function H which is homogeneous of order I is uniquely determined by its restriction 1: on the space C: The function It is said to be a contact Hamiltonian function. We denote the corresponding contact vector field by Xh.