We give a detailed study of the discrete Fourier transform (DFT) of r-even arithmetic functions, which form a subspace of the space of r-periodic arithmetic functions. We consider the DFT of sequences of r-even functions, their mean values and Dirichlet series. Our results generalize properties of the Ramanujan sum. We show that some known properties of r-even functions and of the Ramanujan sum...

An optical implementation of the fractional Fourier transform (FRT) with broadband illumination is proposed by use of a single imaging element, namely, a blazed diffractive lens. The setup displays an achromatized version of the FRT of order P of any two-dimensional input function. This fractional order can be tuned continuously by shifting of the input along the optical axis. Our compact and f...

platinum is a metallic element, which may react with our cellular component through its involvement in cancer chemotherapy medications. cisplatin is one of the most useful antineoplastic drugs against human ovarian carcinoma, which has the central element of platinum in its structure. the nature of chemical interaction between platinum and cellular macromolecules is yet to be understood. we exa...

This chapter covers the fundamental principles and current applications of Fourier transform (FT) infrared and Fourier transform Raman spectroscopies as utilized in the analysis of polymeric materials. The primary emphasis of the first part is on the principles and advantages of these interferometric methods, whereas the remaining sections illustrate numerous applications focusing on the struct...

the knoevenagel condensation of aromatic aldehydes with barbituric acid, dimedone and malononitrile occurred in the presence of bf3/nano-γ-al2o3 at room temperature in ethanol. this catalyst is characterized by powder x-ray diffraction (xrd), fourier transform infrared spectroscopy (ft-ir), thermal gravimetric analysis (tga), field emission scanning electron microscopy (fesem) and energy-disper...