Probabilistic and average linear widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure

Abstract

Let Lq,µ, 1 ≤ q ≤ ∞, denotes the weighted Lq space of functions on the unit ball B d with respect to weight (1 − x 2 2 ) µ− 1 2 , µ ≥ 0, and let W r 2,µ be the weighted Sobolev space on B d with a Gaussian measure ν. We investigate the probabilistic linear (n, δ)-widths λ n,δ (W r 2,µ , ν, Lq,µ) and the p-average linear n-widths λ (a) n (W r 2,µ , µ, Lq,µ)p, and obtain their asymptotic orders for all 1 ≤ q ≤ ∞ and 0 < p < ∞.

Approximation Characteristics of Gel’fand Type in Multivariate Sobolev Spaces with Mixed Derivative Equipped with Gaussian Measure

“…Chen and Fang studied the probabilistic and average widths of multivariate Sobolev space with mixed derivative in the L q -norm, 1 ≤ q < ∞ [13,14]. After 2010, Wang studied the probabilistic and average widths of Sobolev spaces in weighted Sobolev spaces [15,16]. Tan et al studied the Gel'fand N-width in the probabilistic setting of one-dimensional Sobolev space in the L q -norm, 1 ≤ q < ∞ [5].…”

N-Widths of Multivariate Sobolev Spaces with Common Smoothness in Probabilistic and Average Settings in the Sq Norm

“…Dai and Wang [11] obtained the sharp bounds of probabilistic linear (N, δ)-widths and p-average linear N-widths of finite dimensional space with a diagonal matrix. Wang [12,13] estimated the sharp bounds of probabilistic linear (N, δ)-widths and p-average linear N-widths of weighted Sobolev spaces on the ball and Sobolev spaces on compact two-point homogeneous spaces.…”

Fourier orthogonal series on a paraboloid

“…The above narrative turns out to be the prototype for Fourier orthogonal series domains in higher dimensions. In the past two decades, starting from the addition formula for classical orthogonal polynomials on the unit ball [21], closed form formulas for reproducing kernels of orthogonal polynomials have been discovered for several regular domains, including the unit ball, regular simplex, cylinder, as well as the unit sphere with inner product defined by weighted integrals, which makes study of the Fourier orthogonal series on these domain feasible; see, for example, [3,4,5,6,9,10,11,13,18,19,20,21,22] and their references. For unbounded classical domains, we refer to [16] as well as to [2,17] for references on more recent works, which however require techniques beyond our narrative.…”