CONVOLUTION OPERATORS WITH SPLINE KERNELS
| dc.contributor.author | OSMAN, ROHAIZAN | |
| dc.date.accessioned | 2016-01-12T03:45:09Z | |
| dc.date.available | 2016-01-12T03:45:09Z | |
| dc.date.issued | 1994-04 | |
| dc.description.abstract | In this project, we shall derive the asymptotic formulas for convolution operators with spline kernels for higher order differentiable functions. The two classes . of operators which will be considered are the de la Vallee Poussin-Schoenberg operators T m,k with trigonometric B-spline kernel of degree m and the singular integrals of Riemann-Lebesgue Rn,k with the periodic B-spline kernel of degree n-1. These formulas are analogous to the Bernstein's extension of Voronovskaya's estimate for Bernsteins polynomials and Marsden and Riemenschneider's extension of Bernstein-Schoenberg operators for higher order derivatives. In Chapter 1, we shall derive the asymptotic formulas forT m,k by taking limit as mh -+ 0 with m fixed and Rn k as nh -+ 0 with n fixed as well as n -+ co. In ' order to derive these formulas we need to study the asymptotic behaviour of the trigonometric moments of their kernels which can be expressed in terms of their Fourier coefficients and also as polynomials in m and n respectively which can be evaluated usin.g an algorithm. In Chapter 2, we shall derive the asymptotic estimates for T m,k as mh -+ a E (0, 1r] and for Rn,k as nh -+ f3 =/= 0. The former includes the de la Vallee .P oussin means as a special case when a =. 1r. The result for Rn k follows from ' Chapter 1, while for Tm,k, we have to estimate the trigonometric moments for the trigonometric B-spline using its recurrence relation. In Chapter 3, we shall consider discrete bivariate convolution operators t<;> where K is a 2 x 2 nonsingular matrix over 'll, H = 27r K-1 whose range is a space V(K) spanned by {<PK(·- Hn)}nei where I= {no,nl, ... ,n6.-d denotes the representatives of the cosets of 'll2 / K'll2 • vVe shall show that the eigenfunctions Vll of these operators which are independent of a form an orthogonal basis for V(K). We shall also study the limiting behaviour oft~) as JIHII -+ 0 and compute the corresponding limiting semi-groups. The example considered here is the periodic box-spline Bfi:(x), a E (a1,a2,a3,a4)T E JN6 on a 4-directional mesh with a3 = a4 and H is a diagonal matrix. | en_US |
| dc.identifier.uri | http://hdl.handle.net/123456789/1453 | |
| dc.subject | CONVOLUTION OPERATORS | en_US |
| dc.title | CONVOLUTION OPERATORS WITH SPLINE KERNELS | en_US |
| dc.type | Thesis | en_US |
Files
License bundle
1 - 1 of 1
Loading...
- Name:
- license.txt
- Size:
- 1.71 KB
- Format:
- Item-specific license agreed upon to submission
- Description: