Malaysian Journal of Mathematical Sciences, September 2015, Vol. 9, No. 3
Fourier-Like Analysis
Ahmad Safapour and Zohreh Yazdani Fard
Corresponding Email: safapour@vru.ac.ir
Received date: -
Accepted date: -
Abstract:
Fourier transform, its applications and also its generalization is of interests to harmonic analysts. In this paper, we give a new generalization of this transform on $n$-dimensional Euclidean space $\mathbb{R}^{n}$. For this, we define an integral transform on $L^{1}(\mathbb{R}^{n})$ by replacing the ordinary inner product in $n$-dimensional Fourier transform with a quadratic form and prove a
reconstruction formula. The set of all linear operators on $\mathbb{R}^{n}$ preserving this quadratic form with determinant 1 are the indefinite special orthogonal group. We apply this group instead of the ordinary rotation group to perform our new transform. This transform extends to an isometry on $L^{2}(\mathbb{R}^{n})$. The convergence of $n$-dimensional trigonometric Fourier-like series is studied.
Keywords: Fourier-like transform, Integral transform, Quadratic form
