A \(q^ 2\)-analogue operator for \(q^ 2\)-analogue Fourier analysis
The author studies the basic trigonometric functions \(\cos(z;q^2)\) and \(\sin(z;q^2)\) based on the Hahn-Exton \(q\)-Bessel function, which is defined in terms of a \({}_1\phi_1\) basic hypergeometric function with numerator parameter equal to zero. These basic trigonometric functions were introduced by \textit{T. H. Koornwinder} and \textit{R. F. Swarttouw} in [Trans. Am. Math. Soc. 333, No. 1, 445-461 (1992; Zbl 0759.33007)]. The author introduces the \(q\)-difference operator \({\mathcal \delta}_q\) defined by \[ {\mathcal \delta}_q f(z)={f(q^{-1}z)+f(-q^{-1}z)-f(qz)+f(-qz)-2f(-z) \over 2z} \] and proves that \({\mathcal \delta}_q\sin(z;q^2)=\cos(z;q^2)\), \({\mathcal \delta}_q\cos(z;q^2)= -\sin(z;q^2)\) and \({\mathcal \delta}_qe(z;q^2)=e(z;q^2)\), where \(e(z;q^2)= \cos(-iz;q^2)+i\sin(-iz;q^2)\). Finally, the author derives some properties of the \(q\)-Fourier transform defined by the \(q\)-integral \[ \hat{f}(x;q^2)=C\int_{-\infty}^{\infty}f(t)e(-i(1-q)tx;q^2)d_qt, \] where \(C\) is a normalization constant only depending on \(q\).
scientific article; zbMATH DE number 1054003
scientific article