Linear Widths of Weighted Sobolev Classes with Conditions on the Highest Order and Zero Derivatives
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c Pleiades Publishing, Ltd., 2020.
Linear Widths of Weighted Sobolev Classes with Conditions on the Highest Order and Zero Derivatives A. A. Vasil’eva∗,∗∗,1 ∗
Moscow State University, Moscow, 119991 Russia, ∗∗Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia E-mail: 1 vasilyeva [email protected] Received September 2, 2020; Revised September 2, 2020; Accepted September 18, 2020
Abstract. In this paper, order estimates for the linear widths of some function classes are obtained; these classes are defined by restrictions on the weighted Lp1 -norm of the rth derivative and the weighted Lp0 -norm of zero derivatives. DOI 10.1134/S1061920820040135
1. INTRODUCTION In [17], order estimates for the Kolmogorov widths of some weighted Sobolev classes with conditions on the highest order and zero derivatives in a weighted Lebesgue space were obtained. These classes are defined by r ∇ f 1, wf Lp0 (Ω) 1 (1) M = f :Ω→R: g Lp (Ω) 1
(here Ω ⊂ Rd is a domain, r ∈ N, 1 < p0 , p1 ∞, g, w : Ω → (0, ∞) are measurable functions). The weighted Lebesgue space is defined by Lq,v (Ω) = f : Ω → R| f Lq,v (Ω) < ∞ , where f Lq,v (Ω) = f vLq (Ω) ; here 1 q < ∞ and v : Ω → (0, ∞) is a measurable function. Three examples were considered. In the first two of them, Ω was a bounded John domain, the weights were functions of the distance from an h-set Γ ⊂ ∂Ω. In the third example, Ω = Rd , and the weights had the form g(x) = (1 + |x|)β ,
w(x) = (1 + |x|)σ ,
v(x) = (1 + |x|)λ .
(2)
In the present paper, we obtain order estimates for the linear widths of a set M in the space Lq,v (Ω). For p0 = p1 , Mynbaev and Otelbaev [11] obtained upper and lower estimates, but for p1 < 2 < q, they differ in the sense of orders. The problem of estimating the Kolmogorov and linear widths of weighted Sobolev classes with different constraints on the derivatives was studied in [?, ?, ?, ?]. For details, see [17]. Let us give the necessary definitions. Let X be a normed space. By L(X, X) we denote the family of linear continuous operators on X, by rk A, the dimension of the range of an operator A. Let C ⊂ X, n ∈ Z+ . The linear n-width of a set C in the space X is defined as inf sup x − Ax. λn (C, X) = A∈L(X, X), rk An x∈C
For 1 p ∞, we write p =
p p−1 .
˜ θˆ ∈ R, we define the numbers j0 ∈ N and θj ∈ R (1 j j0 ) as follows. Definition 1. Given s∗ , θ, ˜ 1. Let p0 q, p1 q. Then j0 = 2, θ1 = s∗ , θ2 = θ. 2. Let p0 > q, and, in addition, p1 < q 2 or 2 p1 < q. Then j0 = 3, θ1 = s∗ +
1 q
−
1 p1 ,
3. Let p0 q, p1 q, and, in addition, q 2 or min{p0 , p1 } 2. Then j0 = 2, θ1 = s∗ + 537
˜ θ3 = θ. ˆ θ2 = θ, 1 q
−
1 p1 ,
ˆ θ2 = θ.
538
VASIL’EVA
˜ θ3 = θ. ˆ 4. Let p1 > q, and, in addition, p0 < q 2 or 2 p0 < q. Then j0 = 3, θ1 = s∗ , θ2 = θ, 5. Let p0 2 < q, p1 2 < q, ˆ θ3 = θˆ + 1 − 1 , θ4 = qθ . 2
q
1 p0
+ 1q 1,
1 p1
2
6. Let p0 < p1 2 < q,
1 p0
+
1,
1 q
1 p1
+
1 q
1. p (s +1/q−1/p1 ) , 2
(a) If p0 < p1 < 2, then j0 = 5, θ1 = s∗ + 1q − 12 , θ2
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