Estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in integral metrics
T. A. Stepaniuk
备注：
23 pages, in Ukrainian
发表信息：
Zb. Pr. Inst. Mat. NAN Ukr. 11, No 3 (2014), p. 241-269
In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with generating kernel $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k)\cos(kt-\frac{βπ}{2})$, $β\in\mathbb{R}$, whose coefficients $ψ(k)$ are such that product $ψ(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, can't tend to nought faster than every power function and besides, if $1<s<\infty$, then $\sum\limits_{k=1}^{\infty}ψ^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum\limits_{k=1}^{\infty}ψ(k)<\infty$.
Order estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in uniform metric
Order estimation of the best approximations and of the approximations by Fourier sums of classes of $(ψ,β)$--diferentiable functions
通过构造$2π$ -周期连续函数$f$的$C^ψ_{β,p}$类上的三角多项式，建立了最佳一致逼近的正序估计，其中周期连续函数$f$是由属于具有生成固定核$Ψ_β\subset|L_{p'}$，$\frac{1}{p}+\frac{1}{p'}=1$的$L_p$，$1\leq p <\infty$空间中的单位球的函数的卷积定义的，其傅里叶系数以幂函数的形式近似递减到零.在$L_p$ -度量中也建立了精确的阶估计，对于$2π$ -周期函数$f$的$L^ψ_{β,1}$类的$1 < p \leq\infty$，通过Lebesque测度等价于$Ψ_β\subset|L_{p}$核与属于$L_1$空间中单位球的函数的卷积.我们表明，在调查的情况下，最佳逼近的顺序实现的傅立叶和。
