Abstract:A method for time-frequency analysis is given. The approach utilizes properties of Gaussian distribution, properties of Hermite polynomials and Fourier analysis. We begin by the definitions of a set of functions called Harmonic Gaussian Functions. Then these functions are used to define a set of transformations, noted , which associate to a function , of the time variable , a set of functions Ψ which depend on time, frequency and frequency (or time) standard deviation. Some properties of the transformations and the functions Ψ are given. It is proved in particular that the square of the modulus of each function Ψ can be interpreted as a representation of the energy distribution of the signal, represented by the function , in the time-frequency plane for a given value of the frequency (or time) standard deviation. It is also shown that the function can be recovered from the functions Ψ .

We consider the set of powers functions defined on and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.

The main purpose of this work is to identify the general quadratic operator which is invariant under the action of Linear Canonical Transformations (LCTs). LCTs are known in signal processing and optics as the transformations which generalize certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this paper, LCTs corresponding to a general pseudo-Euclidian space are considered. Explicit calculations are performed for the monodimensional case to identify the corresponding LCT invariant operator then multidimensional generalizations of the obtained results are deduced. It was noticed that the introduction of a variance-covariance matrix, of coordinate and momenta operators, and a pseudo-orthogonal representation of LCTs facilitate the identification of the invariant quadratic operator. According to the calculations carried out, the LCT invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of these coordinates and momenta operators themselves. The eigenstates of the LCT invariant operator are also identified with it and some of the main possible applications of the obtained results are discussed.