دانلود رایگان مقاله همگرایی طرح زیربخش غیر ثابت تک متغیره از طریق تشابه مجانبی

Abstract

A new equivalence notion between non-stationary subdivision schemes, termed asymptotic similarity, which is weaker than asymptotic equivalence, is introduced and studied. It is known that asymptotic equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotic equivalence can be relaxed to asymptotic similarity. This result applies to a wide class of non-stationary schemes.

5. Concluding remarks

Non-stationary subdivision schemes are harder to analyze than their stationary counterparts. Analyzing them by comparison with a simpler scheme is quite a natural idea. Up to now, the main tool for such a comparison was the asymptotic equivalence, as developed in Dyn and Levin (1995), see also Dyn et al. (2007). In the present article, we have replaced it by asymptotic similarity, a simpler and weaker equivalence relation between non-stationary schemes. Our work presents a twofold interest. It was clearly important on the theoretical side to point out that asymptotic equivalence may be a too demanding requirement. Indeed, provided that it reproduces constants, a non-stationary scheme which is asymptotically similar to a convergent stationary one is convergent. Certainly, many among the known converging non-stationary schemes are asymptotically equivalent to stationary analogues. Still, we have presented two simple examples which show that asymptotic similarity can also be useful in the design of new converging non-stationary schemes. To enhance the significance of asymptotic similarity, we would like to mention that this notion was first introduced for non-regular (non-uniform, non-stationary) schemes in Mazure (in press) (simply named there equivalence). However, the non-uniformity made it necessary to define it locally. In this non-regular framework, it proved to be a forceful tool to analyze not only the convergence of a subdivision scheme but also the regularity of the limit functions it produces. As pointed out by D. Levin, in the uniform setting, smoothness of the generated limits can be proved by combining asymptotic similarity with the notion of smoothing factors introduced in Dyn and Levin (1995). Further results concerning asymptotic similarity for uniform non-stationary schemes can be found in Charina et al. (2015) along with several important extensions of this notion.