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Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by exp(-x 2 /2), which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials