Cosine Manifestations of the Gelfand Transform

The Discrete Cosine Transform

Each Discrete Cosine Transform uses N real basis vectors whose components are cosines. In the DCT-4, for example, the jth component of v k is cos(j + 1 2)(k + 1 2) N. These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the intensities along a row of pixels, its cosine series P c k v k has the coeecients c k = (x; v k)=N. They are q...

JPEG Encoder using Discrete Cosine Transform & Inverse Discrete Cosine Transform

In the past decade, the advancement in data communications was significant during explosive growth of the Internet, which led to the demand for using multimedia in portable devices. Video and Audio data streams require a huge amount of bandwidth to be transferred in an uncompressed form. The objective of this paper is to minimize the number of bits required to represent an image and also the ac...

The Gelfand Transform in Commutative Algebra

We consider the transformation ev which associates to any element in a K-algebra A a function on the the set of its K-points. This is the analogue of the fundamental Gelfand transform. Both ev and its dual ev∗ are the maps from a discrete K-module to a topological Kmodule and we investigate in which case the image of each map is dense. This question arises in the classical problem of the recons...

Projective Hulls and the Projective Gelfand Transform

We introduce the notion of a projective hull for subsets of complex projective varieties parallel to the idea of a polynomial hull in affine varieties. With this concept, a generalization of J. Wermer’s classical theorem on the hull of a curve in Cn is established in the projective setting. The projective hull is shown to have interesting properties and is related to various extremal functions ...

Discrete Cosine Transform