Diverse Smooth Function Of Class Width And Optimal Recovery

This dissertation includes three chapters. In chapter 1, we study the approximation of multivariate Sobolev classes and multivariate Besov classes by multivariate polynomial splines and the optimal recovery of multivariate Sobolev classes and multivariate Besov classes. In chapter 2, we study the approximation of multivariate Sobolev classes and multivariate Besov classes by multivariate trigonometric polynomial splines and the approximation of multivariate Sobolev classes and multivariate Besov classes by the exponential type integral functions. In chapter 3, we discuss the n-widths of Generalized Besov classes in the Sobolev classes.Let (X(R^d), || ? ||x) be a normed space of real functions on R^d. Let α > 0 and P_α be the operator in X(R^d) defined by P_αf(x) = X_α(x)f(x) (where X_α(x) is the characteristic function of the cube I_α^d = [-α,α]^d ). Let L be a subspace of X(R^d). Set P_αL = {P_αf : f ∈ L}. Suppose L is locally-finite dimensional, i.e., dim(P_αL,X(R^d)) < +∞ for every α > 0. Then the following quantity is said to be the average dimension of L in X (in the sense of LFD).Let σ > 0, and C be a centrally symmetric subset of X(R^d). The infinite-dimensional Kolmogorov σ-width of C in X(R^d) is defined to bewhere the infimum is taken over all L with [(dim)|～](L,X(R^d)) ≤ σ.The infinite-dimensional linear σ-width of C in X(R^d) is defined to bewhere the infimum is taken over all A which are continuous linear operators from span(C) to X(R^d) and [(dim)|～](Im∧, X(R^d)) ≤ σ, Im∧ denotes the range of the operator ∧. The infinite-dimensional Bernstein σ-width of C in X(R^d) is defined to beb_σ(C,X(R^d)) := sup sup{λ > 0 | L∩λBX(R^d) (?) C},where the supremum is taken over all L with [(dim)|～] (L, X(R^d)) ≤ a and d_σ (L∩BX (R^d), X (R^d))= 1, and BX(R^d) denote the unit ball in X(R^d).Let L be a subspace of X(R^d), ε > 0, andk_ε(α,L,X(R^d)) := min{n ∈ Z₊|d_n(P_α(L∩BX(R^d)),X(R^d)) < ε}.The quantityis said to be the average dimension of L in X.The average Kolmogorov σ-width, the average linear σ-width, and the average Bernstein σ-width of C in X(R^d) are defined similarly (see [9]). and we denote them by d_σ(C,X(R^d)), 3_σ(C,X(R^d)), and b_σ(C,X(R^d)), respectively.For 1 ≤ p, g ≤ ∞, denote by L_pq(R^d) the linear normed space of locally L_p-integrable functions f with the following finite normFor 1 ≤ p, q ≤ ∞, r ∈ N, the isotropic Sobolev-Wiener classes W_pq^r(R^d) are defined to beFor 1 ≤ p, q ≤ ∞,r = (r₁, … ,r_d) ∈ N^d,M = (M₁,…, M_d) ∈ R₊^d, the anisotropic Sobolev-Wiener classes W_pq^r(M, R^d) are defined to beFor σ > 1, let ρ = 1, … ,d. The multivariate polynomial spline space S_σ,r-1(R^d) is defined to beLet ∏_n denote the collection of all polynomials of degree not exceeding n. Denote by L_n(x) the cardinal spline satisfying conditions:1) L_n(x) ∈ S_n = {s:s∈ C^n-1(R),s|(v,v+1) ∈ ∏_n,(?) ∈ z},2) L_n(α_n + v) = δ_αv, v ∈ Z, where α_n = (1 + (-l)^n-1)/43) |L_n(x)| ≤ Ae^-B|x| x ∈ R, for some positive constants A and B.For j = where Liu Yongping [29], Luo Junbo, Liu Yongping [35] discussed the average Kolmogorov σ-width of Sobolev-Wiener classes. When d=1, Magaril-Il'yaev [9], Li Chun [42], etc. discussed the approximation of Sobolev classes by the cardinal spline interpolation operator, and obtained the infinite-dimensional σ-widths of Sobolev classes. In this paper. We obtainTheorem 1.1 Let 1 ≤ q ≤ p ≤ ∞. σ > 1. ThenDenote by X*(R^d) the dual space X(R^d). The average codimension can be different values for the same subspace according to the definition of Magaril-Il'yaev G.G. [9] and the functions what he used G(f) = f(x_j), (?)f ∈ L_p(R^d) does not belong to L_p^*(R^d) for 1 ≤ p ≤ ∞. To improve it, professor Sun Yongsheng give the following definition of the infinite-dimensional Gel'fand σ-width.Let X*(R^d) also be a normed linear space on R^d. For σ > 0, we take the annulling class R_σ which consists of all linear subspace of X*(R^d) with the locally finite dimensional property, and moreover, f...