Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions
نویسندگان
چکیده
Abstract. Let I be a finite interval, r, n ∈ N, s ∈ N0 and 1 ≤ p ≤ ∞. Given a set M , of functions defined on I, denote by ∆+M the subset of all functions y ∈ M such that the s-difference ∆τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by W r p the Sobolev class of functions x on I with the seminorm ‖x‖Lp ≤ 1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes ∆+W r p in Lq for s > r + 1 and (r, p, q) 6= (1, 1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically 3 n−2.
منابع مشابه
SHAPE PRESERVING WIDTHS OF SOBOLEV-TYPE CLASSES OF s-MONOTONE FUNCTIONS ON A FINITE INTERVAL
Let I be a finite interval and r ∈ N. Denote by ∆ s + L q the subset of all functions y ∈ L q such that the s-difference ∆ s τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by ∆ s + W r p , the class of functions x on I with the seminorm x (r) L p ≤ 1, such that ∆ s τ x ≥ 0, τ > 0. For s = 3,. .. , r + 1, we obtain two-sided estimates of the shape preserving widths
متن کاملShape Preserving Widths of Weighted Sobolev-type Classes of Positive, Monotone and Convex Functions on a Finite Interval
Let X be a real linear space of vectors x with a norm ‖x‖X , W ⊂ X, W 6= ∅ and V ⊂ X, V 6= ∅. Let L be a subspace in X of dimension dim L ≤ n, n ≥ 0 and M = M(z) := z + L be a shift of the subspace L by an arbitrary vector z ∈ X. If M ∩ V 6= ∅, then we denote by E(x, M ∩ V )X := inf y∈Mn∩V ‖x− y‖X , the best approximation of the vector x ∈ X by M ∩ V , and by E(W,M ∩ V )X := sup x∈W E(x,M ∩ V )X ,
متن کاملOptimal spline spaces of higher degree for L n-widths
In this paper we derive optimal subspaces for Kolmogorov n-widths in the L2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases. Math Subject C...
متن کاملOptimal spline spaces of higher degree for L2 n-widths
In this paper we derive optimal subspaces for Kolmogorov n-widths in the L2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases. Math Subject C...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Journal of Approximation Theory
دوره 140 شماره
صفحات -
تاریخ انتشار 2006