INTEGRAL REPRESENTATIONS FOR THE DERIVATION OF SAMPLING EXPANSIONS.
PRINCETON UNIV N J DEPT OF ELECTRICAL ENGINEERING
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The work presents an integral representation for the derivation of sampling expansions. The representation is based on a property of the resolvent operator on a Hilbert space. The theory of formally self-adjoint differential operators and boundary-value problems is used to obtain a more explicit expression for the integral representation in terms of the Greens function. The principal interest of this representation is the different physical significanes of the permutations possible in performing the integral. Two permutations yield two commonly used methods for the derivation of sampling expansions and offer interesting interpretations in terms of physical systems. The first order differential operator is considered with non-self-adjoint boundary conditions. The representations for this case yield periodically nonuniform and derivative sampling expansions, with corresponding system interpretation. Then the second order differential operator is considered. The general forms of the interpolating functions are found. Examples resulting in nonuniform sampling expansions are discussed. Expansions in Gegenbauer polynomials and Bessel functions are given to illustrate the validity of the method for singular cases as well. Author
- Theoretical Mathematics