Let $H=C^{r,\alpha }([0,1{]}^d)$ be H$\ddot{\text{o}}$lder space and $G=L_2([0,1{]}^d)$ with the inner product given by $$⟨g,h{⟩}_G={\int }_{[0,1{]}^d}g(x)h(x)dx\mathrm{\forall }g,h\in G.$$ This paper considers the embedding operator $S:H\to G,S(f)=f,f\in H$. We prove that $$e_n(S,{\mathit{\Lambda }}^{\text{std}})\le \underset{k=0,1,\cdots }{min}{(e_k(S,{\mathit{\Lambda }}^{\text{all}}{)}^2+\frac{C\cdot k}{n\cdot n^{\frac{2(r+\alpha )}{d}}})}^{1/2},$$ where $e_n(S,{\mathit{\Lambda }}^{\text{std}})$ and $e_n(S,{\mathit{\Lambda }}^{\text{all}})$ denote the \textit{n}th minimal error of standard and linear information respectively in the worst case, average case and randomized settings, and $C$ is a constant.

Let $H=C^{r,\alpha }([0,1{]}^d)$ be H$\ddot{\text{o}}$lder space and $G=L_2([0,1{]}^d)$ with the inner product given by $$⟨g,h{⟩}_G={\int }_{[0,1{]}^d}g(x)h(x)dx\mathrm{\forall }g,h\in G.$$ This paper considers the embedding operator $S:H\to G,S(f)=f,f\in H$. We prove that $$e_n(S,{\mathit{\Lambda }}^{\text{std}})\le \underset{k=0,1,\cdots }{min}{(e_k(S,{\mathit{\Lambda }}^{\text{all}}{)}^2+\frac{C\cdot k}{n\cdot n^{\frac{2(r+\alpha )}{d}}})}^{1/2},$$ where $e_n(S,{\mathit{\Lambda }}^{\text{std}})$ and $e_n(S,{\mathit{\Lambda }}^{\text{all}})$ denote the \textit{n}th minimal error of standard and linear information respectively in the worst case, average case and randomized settings, and $C$ is a constant.