[2] Apparently Fourier did not possess the inner-product expression for (concrete) Fourier coefficients when he first proposed that functions are expressible in Fourier series. This understandably weakened his claim. Worse, at the time, notions of convergence and even the notion of function were amorphous, so that the sense(s) in which a function might be represented by a Fourier series could n...

This paper review history of work in the development of abstract harmonic analysis, which is a mathematical discipline attempting to extend the classical Fourier analysis to various groups.

We give an overview of phylogenetic invariants: a technique for reconstructing evolutionary family trees from DNA sequence data. This method is useful in practice and is based on a number of simple ideas from elementary group theory, probability, linear algebra, and commutative algebra.

This summary of the doctoral thesis [8] is created to emphasize the close connection of the proposed spectral analysis method with the Discrete Fourier Transform (DFT), the most extensively studied and frequently used approach in the history of signal processing. It is shown that in a typical application case, where uniform data readings are transformed to the same number of uniformly spaced fr...

The class of second-order curvilinear gratings consists of all the curvilinear gratings that are obtained by second-order spatial transformations of periodic gratings. It includes, for example, circular, elliptic, and hyperbolic gratings as well as circular, elliptic, and hyperbolic zone plates. Such structures occur quite frequently in optics, and their Fourier transforms may arise, for instan...

We give a discussion of harmonic analysis on finite abelian groups, emphasizing various ways in which the structure of the group is encoded on various spaces of functions, ways in which the Fourier transform detects and preserves these structures. We discuss the major tools, like convolutions and Fourier transforms, along with some fundamental theorems, like the Plancheral, Parseval, Fourier in...

The following paper presents the results of tests conducted by the author in the field of verification of influence of JPEG lossy compression on the robustness of data hiding methods used for the colour image which has been transformed to the frequency domain. The data have been embedded to the Fourier magnitude spectrum of the image. The possibility of decoding the hidden data with two embeddi...

Article history: Received 25 June 2014 Received in revised form 8 July 2014 Accepted 10 August May 2014 Available online 30 August 2014

How can one prove a sharp inequality? Symmetrization, Fourier analysis, and probability are often used, and we will survey some of these methods through examples. We then survey sharp constants in Grothendieck inequalities, leading to some recent work on computing the best constant for a Grothendieck-type inequality of Khot and Naor. (Joint work with Aukosh Jagannath and Assaf Naor)

Both theorems, (I) and the stronger (II), although they are very simple in their statements, have been conjectured to require Fourier analysis for their proofs; see, e.g., p. 396 of Billingsley [2] for the first and p. 49 of Billingsley [1] for the second part. This note gives probabilistic proofs of the two theorems and thus answers this conjecture to the negative. The main argument of the pro...