设$\{u_{m}(t)\}_{m=0}^{\infty}$为第二类切比晓夫多项式，$\{\varphi_{m}\}_{m=0}^{\infty}$为任意实数列. 本文主要考虑了单位圆盘$U$上的切比晓夫脊多项式 系统$\{\{u_{m}(x\cos(\frac{k\pi}{m+1}+\varphi_{m})+y\sin(\frac{k\pi}{m+1}+\varphi_{m})\}_{k=0}^{m} \}_{m=0}^{\infty}$在$L_{p}(U) (1\leqslant p \leqslant \infty$，当$p= \infty$时，为$C(U))$空间 上Chebyshev-Fourier系数的敛散问题和单位圆盘$U$上的可微函数类在$L_{2}$尺度下的相对宽度问题.\bigskip \par 文章主要分为两个部分，第一部分讨论了在$L_{p},1\leqslant p < \frac{3}{2}$上Chebyshev-Fourier系数的 敛散情况，得到了定理1. 对所有实数$p, 1 \leqslant p < \frac{3}{2}$，存在一个函数$g\in L_{p}(U)$，使得 \begin{equation*}\ \omega_{1}(g, \delta)_{p}=O(\delta^{\frac{1}{p}-\frac{2}{3}}) \quad (\delta\rightarrow 0+), \quad \omega_{2}(g, \delta)_{p}=0, \end{equation*} 并且对于任意实序列$\{\varphi_{m}\}_{m=0}^{\infty}$成立 \begin{equation*} \label{14} \varlimsup\limits_{m\rightarrow\infty}\max\limits_{0\leqslant k \leqslant m}|a_{m}(g,k,\varphi_{m})|\geqslant C >0, \end{equation*} 其中$C$为常数，$a_{m}(g,k,\varphi_{m})$为相对于切比晓夫脊多项式 系统的Chebyshev-Fourier系数.进而得到了推论2，即此时Chebyshev-Fourier系数不收敛于零，从而解决了圆盘上切比晓夫 脊多项式系统Chebyshev-Fourier系数的敛散性问题.\par

Let $u_{m}(t)$ be the Chebyshev polynomials of the second kind,$\{\varphi_{m}\}_{m=0}^{\infty}$ be an arbitrary real sequence and$\{\{u_{m}(x\cos(\frac{k\pi}{m+1}+\varphi_{m})+y\sin(\frac{k\pi}{m+1}+\varphi_{m})\}_{k=0}^{m} \}_{m=0}^{\infty}$ be the systems of Chebyshev ridge polynomials on unit disc $U$. This paper mainly discusses the convergence and divergence of its Chebyshev-Fourier coefficients on the space of $L_{p}(U) (1\leqslant p \leqslant \infty$，if $p= \infty$，then $C(U))$, and the relative widths of classes of differentiable functions in the space $L_{2}(U)$.\bigskip \par The paper contains two main parts, the first one discusses the convergence of the Chebyshev-Fourier coefficients on the space of $L_{p},1\leqslant p < \frac{3}{2}$, and obtains the theorem 1. For all real number $p, 1 \leqslant p < \frac{3}{2}$, there exists a function $g\in L_{p}(U)$ such that $$\omega_{1}(g, \delta)_{p}=O(\delta^{\frac{1}{p}-\frac{2}{3}}),(\delta\rightarrow 0+), \quad \omega_{2}(g, \delta)_{p}=0. $$ Moreover, for all real sequence$\{\varphi_{m}\}_{m=0}^{\infty}$ hold $$\varlimsup\limits_{m\rightarrow\infty}\max\limits_{0\leqslant k \leqslant m} |a_{m}(g,k,\varphi_{m})|\geqslant C >0,$$ where $C$ is a constant. Then we get the corollary 2, and solve the problem.\par The second part discusses the relative widths of classes of differentiable