Let $C^{m,\alpha}_M([0,1])$ be a Holder ball consisting of real-valued functions $g$ on $[0,1]$ such that $$ \|g\|_{C^{m,\alpha}} := \max_{0\leq j \leq m } \sup_{x\in [0,1]} |g^{(j)}(x)| + \sup_{x,y\in [0,1], x\neq y} \frac{ |g^{(m)}(x) -g^{(m)}(y)|}{|x-y|} \leq M. $$ Let $f \in C^{m,\alpha}_M([0,1])$ be fixed.

By Weierstrass approximation theorem, there exists a sequence of polynomials $$p_n(x) = \sum_{j=0}^{J_n} \alpha_{n,j} x^j$$ such that $\|f-p_n\|_{\infty} \to 0$ as $n \to \infty$. Here $J_n \to \infty$.

Questions:

Does there exist approximating polynomials $p_n \in C^{m,\alpha}_M([0,1])$, such that $\|f-p_n\| \to 0$? If this is not true in general, for what kind of functions $f$ it may hold?

Motivation of these questions:

I am reading some papers in nonparametric statistics, Chen (2007) and Chen and Ai (2003). They rely on approximating an unknown function in some space, i.e. $f\in C_M^{m,\alpha}$ here, using functions in some approximating spaces increasing with $n$, i.e. space of $J_n$-th order polynomials here.

To have desirable statistical properties, a key assumption they used is that, the approximating spaces are subsets of the original space $C^{m,\alpha}_M([0,1])$. Their practice corresponds to the question. I am not sure whether adding such further restriction on the approximating spaces will cause serious issues on the approximating ability. This issue is not discussed in the listed papers, nor in other related papers I have checked.

Any reference or discussions is greatly appreciated!

Let $C^{m,\alpha}_M([0,1])$ be a Holder ball consisting of real-valued functions $g$ on $[0,1]$ such that $$ \|g\|_{C^{m,\alpha}} := \max_{0\leq j \leq m } \sup_{x\in [0,1]} |g^{(j)}(x)| + \sup_{x,y\in [0,1], x\neq y} \frac{ |g^{(m)}(x) -g^{(m)}(y)|}{|x-y|^\alpha} \leq M. $$ Let $f \in C^{m,\alpha}_M([0,1])$ be fixed.

By Weierstrass approximation theorem, there exists a sequence of polynomials $$p_n(x) = \sum_{j=0}^{J_n} \alpha_{n,j} x^j$$ such that $\|f-p_n\|_{\infty} \to 0$ as $n \to \infty$, where $\|\cdot\|_\infty$ denotes the uniform norm. Here $J_n \to \infty$.

Question:

Does there exist approximating polynomials $p_n \in C^{m,\alpha}_{M'}([0,1])$ for some $M'>0$, such that $\|f-p_n\|_{\infty} \to 0$? If this is not true in general, for what kind of functions $f$ it may hold?

Motivation of the question:

I am reading some papers in nonparametric statistics, Chen (2007) and Chen and Ai (2003). They rely on approximating an unknown function in some space, i.e. $f\in C_M^{m,\alpha}$ here, using functions in some approximating spaces increasing with $n$, i.e. space of $J_n$-th order polynomials here.

To have desirable statistical properties, a key assumption they used is that, the approximating spaces are subsets of the original space $C^{m,\alpha}_M([0,1])$. Their practice corresponds to the question. I am not sure whether adding such further restriction on the approximating spaces will cause serious issues on the approximating ability. This issue is not discussed in the listed papers, nor in other related papers I have checked.

Any reference or discussions is greatly appreciated!

Let $C^{m,\alpha}_M([0,1])$ be a Holder ball consisting of real-valued functions $g$ on $[0,1]$ such that $$ \|g\|_{C^{m,\alpha}} := \max_{0\leq j \leq m } \sup_{x\in [0,1]} |g^{(j)}(x)| + \sup_{x,y\in [0,1], x\neq y} \frac{ |g^{(m)}(x) -g^{(m)}(y)|}{|x-y|} \leq M. $$ Let $f \in C^{m,\alpha}_M([0,1])$ be fixed.

By Weierstrass approximation theorem, there exists a sequence of polynomials $$p_n(x) = \sum_{j=0}^{J_n} \alpha_{n,j} x^j$$ such that $\|f-p_n\|_{\infty} \to 0$ as $n \to \infty$. Here $J_n \to \infty$.

Questions:

1. Does there exist approximating polynomials $p_n \in C^{m,\alpha}_M([0,1])$, such that $\|f-p_n\| \to 0$? If this is not true in general, for what kind of functions $f$ it may hold?

2. If restricting the polynomial coefficients $\sum_{j=0}^{J_n} |\alpha_{n,j}| <M_f$ for some fixed large constant $M_f$ independent of $n$, does there exist polynomials $p_n$ such that $\|f-p_n\| \to 0$? I can see this is true for some specific functions, e.g. $f(x)=e^x$ with $M_f > e$. But does this hold for a general $f$? If not, for what kind of functions $f$ it may hold?

3. What if we use different types of approximating functions, e.g. using Fourier basis instead of polynomials?

Motivation of these questions:

I am reading some papers in nonparametric statistics, Chen (2007) and Chen and Ai (2003). They rely on approximating an unknown function in some space, i.e. $f\in C_M^{m,\alpha}$ here, using functions in some approximating spaces increasing with $n$, i.e. space of $J_n$-th order polynomials here.

To have desirable statistical properties, a key assumption they used is that, the approximating spaces are subsets of the original space $C^{m,\alpha}_M([0,1])$. Their practice corresponds to Question 1. I am not sure whether adding such further restriction on the approximating spaces will cause serious issues on the approximating ability. This issue is not discussed in the listed papers, nor in other related papers I have checked.

Any reference or discussions is greatly appreciated!

Let $C^{m,\alpha}_M([0,1])$ be a Holder ball consisting of real-valued functions $g$ on $[0,1]$ such that $$ \|g\|_{C^{m,\alpha}} := \max_{0\leq j \leq m } \sup_{x\in [0,1]} |g^{(j)}(x)| + \sup_{x,y\in [0,1], x\neq y} \frac{ |g^{(m)}(x) -g^{(m)}(y)|}{|x-y|} \leq M. $$ Let $f \in C^{m,\alpha}_M([0,1])$ be fixed.

By Weierstrass approximation theorem, there exists a sequence of polynomials $$p_n(x) = \sum_{j=0}^{J_n} \alpha_{n,j} x^j$$ such that $\|f-p_n\|_{\infty} \to 0$ as $n \to \infty$. Here $J_n \to \infty$.

Questions:

Does there exist approximating polynomials $p_n \in C^{m,\alpha}_M([0,1])$, such that $\|f-p_n\| \to 0$? If this is not true in general, for what kind of functions $f$ it may hold?

Motivation of these questions:

I am reading some papers in nonparametric statistics, Chen (2007) and Chen and Ai (2003). They rely on approximating an unknown function in some space, i.e. $f\in C_M^{m,\alpha}$ here, using functions in some approximating spaces increasing with $n$, i.e. space of $J_n$-th order polynomials here.

To have desirable statistical properties, a key assumption they used is that, the approximating spaces are subsets of the original space $C^{m,\alpha}_M([0,1])$. Their practice corresponds to the question. I am not sure whether adding such further restriction on the approximating spaces will cause serious issues on the approximating ability. This issue is not discussed in the listed papers, nor in other related papers I have checked.

Any reference or discussions is greatly appreciated!