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I'm struggling either proving or disproving the following statement:

Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If $S$ is dense in $C(K)$$C^1(K)$ with respect to sup-norm, then for any $f\in C(K)$$f\in C^1(K)$, there exists $\{g_n\} \subset S$ such that $g_n$ uniformly approximates $f$ with $f'$ uniformly approximated by $g_n'$.

This statement seems to be true, although $S$ is only a subset of all polynomials over $K$. If the statement appears false, is there any prior assumption that makes it true? Any reference or direct answer would greatly help me.

I'm struggling either proving or disproving the following statement:

Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If $S$ is dense in $C(K)$ with respect to sup-norm, then for any $f\in C(K)$, there exists $\{g_n\} \subset S$ such that $g_n$ uniformly approximates $f$ with $f'$ uniformly approximated by $g_n'$.

This statement seems to be true, although $S$ is only a subset of all polynomials over $K$. If the statement appears false, is there any prior assumption that makes it true? Any reference or direct answer would greatly help me.

I'm struggling either proving or disproving the following statement:

Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If $S$ is dense in $C^1(K)$ with respect to sup-norm, then for any $f\in C^1(K)$, there exists $\{g_n\} \subset S$ such that $g_n$ uniformly approximates $f$ with $f'$ uniformly approximated by $g_n'$.

This statement seems to be true, although $S$ is only a subset of all polynomials over $K$. If the statement appears false, is there any prior assumption that makes it true? Any reference or direct answer would greatly help me.

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mw19930312
mw19930312
  • 263
  • 1
  • 6

Uniformly approximating a function and its derivative using polynomials

I'm struggling either proving or disproving the following statement:

Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If $S$ is dense in $C(K)$ with respect to sup-norm, then for any $f\in C(K)$, there exists $\{g_n\} \subset S$ such that $g_n$ uniformly approximates $f$ with $f'$ uniformly approximated by $g_n'$.

This statement seems to be true, although $S$ is only a subset of all polynomials over $K$. If the statement appears false, is there any prior assumption that makes it true? Any reference or direct answer would greatly help me.