二元连续周期函数用其Fourier级数的Marcinkiewicz型线性平均逼近

ON THE APPROXIMATION OF CONTINUOUS PERIODIC FUNCTIONS OF TWO VARIABLES BY ITS LINEAR MEANS OF MARCINKIEWICZ TYPE

§1.引言用 C2n×2n表示对每个变元均以2π为周期的二元连续函数空间,其范数为‖f‖c=■|f（x,y）|。函数类 Hω1,ω2。由 C2n×2n中满足下述条件的函数 f（x。

Let ω_1,ω_2 be convex modulus of continuity and satisfy the condition: ■δ∫_δ~1(ω_i(t))/(t^2)dt=O(ω_i(δ))(i=1,2)δ∈(0,1), H^(ω_1,ω_2) be the class of 2π-periodic continuous functions f(x,y)such that |f(x,y)-f(x″,y′)|≤ω_1(|x-x′|)+ω_2(|y-y′|) ■x,x′,y,y′. Define the linear means Ln(f;x,y) of Marcinkiewicz type by L_n(f;x,y)=■(λ_k^((n)))-λ_(k=1)^((n))S_(k,k)(f;x,y), where S_(k,k)(f) is the partial sum of Fourier series of f.If ■ satisfies the following conditions: (1)■; (2)■ is a convex or concave sequence; (3)-1λ_1^((n))=O(1/n), then ■