We consider Jackson inequality in L^2(B^d × T,W_(κ,μ)~B(x)),where the weight function W_(κ,μ)~B(x) is defined on the ball B^d and related to reflection group,and obtain the sharp Jackson inequality E_(n- 1,m-1)(f)2 ≤k_(n,m)(r,r)ω_r(f,t)_2,T ≥ 2τ_(n,λ),where τ_(n,λ) is the first positive zero of the Gegenbauer cosine polynomial C_n~λ(cosθ)(n ∈ N).
In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L2,κ(Rd) by a subspace E2κ(σ)(SE2κ(σ)), which is a subspace of entire functions of exponential type(spherical exponential type) at most σ. Here L2,κ(Rd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight v2κ(x) =ξ∈R, which is defined by a positive+|(ξ, x)|κ(ξ)subsystem R+ of a finite root system RRdand a function κ(ξ) : R → R+ invariant under the reflection group G(R) generated by R. In the case G(R) = Zd2, we get some exact results. Moreover,the deviation of best approximation by the subspace E2κ(σ)(SE2κ(σ)) of some class of the smooth functions in the space L2,κ(Rd) is obtained.
In this paper,we consider the best EFET(entire functions of the exponential type) approximations of some convolution classes associated with Laplace operator on R d and obtain exact constants in the spaces L1(R2) and L2(Rd).Moreover,the best constants of trigonometric approximations of their analogies on Td are also gained.
