The paper provides a full self-contained derivation of fast algorithms to compute discrete Cosine and Sine transforms I IV. For the Sine I/II and Cosine I/II transforms a unified derivation based on the concept of the comrade matrix is presented. The comrade matrices associated with different versions of the transforms differ in only a few boundary elements; hence, in each case algorithms can b...

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. The two identities are intimately connected with the classical circle and divisor problems, respectively. For each of Ramanujan’s identities, there are three possible interpretations for the double series. In two earlier papers, t...

We have studied the solutions of the combined sine–cosine-Gordon Equation found by Wazwaz [A.M. Wazwaz, Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method, Appl. Math. Comput. 177 (2006) 755] using the variable separated ODE method. These solutions can be transformed into a new form. We have derived the relation between t...

The proposed contribution describes a method for the detection and classification of harmonic transients, as they occur in electrical networks. The method uses, as bases, a special class of wavelet-packets, i. e. smooth biorthogonal sine and cosine-packets in order to find the best ’’Shannon-basis“. The algorithm is based on the minimization of Shannon‘s function for the decomposed wanted signa...

FRACTIONAL FOURIER INTEGRAL THEOREM AND FRACTIONAL FOURIER SINE AND COSINE TRANSFORM Saleem Iqbal, S.M. Raza, * LalaRukh Kamal and Farhana Sarwar Department of Mathematics/Physics, University of Balochistan, Quetta, Pakistan e-mail: fs1005,saleemiqbal81,[email protected]. ABSTRACT: The fractional Fourier transform (FRFT) is a generalization of the ordinary Fourier transform (FT). Recently...

In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. In this paper, we will introduce sev...

In this paper we will discuss the usefulness of overcomplete systems of basis functions, such as wavelets or localized sine and cosine functions, for the adaptive parsimonious representation and estimation of statistical signals which show an inhomogeneous behaviour over time. Typical examples can be found in electrical engineering, sound and speech processing, geophysics, biomedicine, etc. We ...

We will investigate the superstability of the hyperbolic trigonometric functional equation from the following functional equations: fx y ± gx − y λfxgy andfx y ± gx − y λgxfy, which can be considered the mixed functional equations of the sine function and cosine function, the hyperbolic sine function and hyperbolic cosine function, and the exponential functions, respectively.