Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

Fourier transforms of dini-lipschitz functions

Abstract: It is well known that if Lipschitz conditions of a certain order are imposed on a function f(x), then these conditions affect considerably the absolute convergence of the Fourier series and Fourier transforms of f. In general, if f(x) belongs to a certain function class, then the Lipschitz conditions have bearing as to the dual space to which the Fourier coefficients and transforms of f(x) belong. In the present work we do study the same phenomena for the wider Dini-Lipschitz class as well as for some other allied classes of functions.

Eisenstein series for SL(2)

Abstract: This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local Whittaker functions over SL2. They are used to compute both the value and derivatives of these functions at critical points.

New efficient Fourier-reconstruction method for approximate image reconstruction in spiral cone-beam CT at small cone angles

Abstract: We present an approximate algorithm for image reconstruction in spiral cone-beam CT at small cone-angles. The first step of the algorithm is a rebinning from fanbeam to parallel beam projections, that is performed independently for each detector row. The result of this rebinning procedure is a set of parallel views, where all rays are tilted against the z-axis by their cone angle and the rays within each view have different z-positions. After that, as with most of the approximate algorithms, the basic idea is to let each oblique ray contribute to the image with a weight that depends on its distance to the image plane. Because the distance of a ray to the image plane changes along the ray, so does its weight. This is the one important difference compared to standard 2D reconstruction procedures. However, for small cone angles, the variation of the weight along a ray is rather smooth, so that we can synthesize it as a Fourier series using only few Fourier coefficients. In standard 2D image reconstruction, we can build the 2D Fourier spectrum of the image on a polar grid from the Fourier transforms of the views, where each view contributes to a radial line only, because the weights are constant along the rays. In our algorithm, the weights change along the rays. This variation is modeled as a Fourier series with 2N(mu ) + 1 coefficients. Hence, the views contribute not only to one but to 2N(mu ) + 1 lines in the 2D Fourier space of the image. For a given pitch, the Fourier coefficients that take into account the variations of the weights are precalculated before the reconstruction and stored in a table. From the generated 2D image spectrum, we calculate the final image using a gridding technique to convert the sampling grid to cartesian and finally doing a 2D IFFT. We have evaluated our method using simulated test data for various test cases assuming sampling conditions and image quality requirements typical to medical CT. These experiments have shown that the method is applicable for detector arrays with up to at least 32 rows.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.