Biorthogonal exponential sequences with weight function exp(𝑎𝑥²+𝑖𝑏𝑥) on the real line and an orthogonal sequence of trigonometric functions

Masjed-Jamei, Mohammad

doi:10.1090/S0002-9939-07-09139-3

Biorthogonal exponential sequences with weight function $\exp (ax^2+ibx)$ on the real line and an orthogonal sequence of trigonometric functions
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Proc. Amer. Math. Soc. 136 (2008), 409-417 Request permission

Abstract:

Some orthogonal functions can be mapped onto other orthogonal functions by the Fourier transform. In this paper, by using the Fourier transform of Stieltjes–Wigert polynomials, we derive a sequence of exponential functions that are biorthogonal with respect to a complex weight function like $\exp (q_1(ix+p_1)^2+q_2(ix+p_2)^2)$ on $(-\infty ,\infty )$. Then we restrict these introduced biorthogonal functions to a special case to obtain a sequence of trigonometric functions orthogonal with respect to the real weight function $\exp (-qx^2)$ on $(-\infty ,\infty )$.

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