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I am quite confident that the answer to the first question is no (for this exact version):

I think you need to be careful the relativity of your assumptions at least in the first case. Consider two sequences of functions $\hat{f}_n$ with $|\hat{f}_n(x)| < C$ and $\hat{g}_n(x)$ with $c < |\hat{g}_n| < C$ for all $n \in \mathbb{N}$ and $x \in \mathbb{R}^d$.

Now define $f_n = \frac{1}{n^2} \hat{f}_n$ and $g_n = \frac{1}{n} \hat{g}_n$. For sure, $f_n, g_n \rightarrow 0$ and $$|\frac{f_n}{g_n}| < \frac{C}{nc} \rightarrow 0.$$ At the same time, this rescaling does not change minimizers, so $f_n/g_n$ has the same as $\hat{f}_n/\hat{g}_n$. The only condition however that remains with the hat functions is boundedness.

Now you can surely construct two sequences of hat functions that are bounded, but for which the influence of $g$ does not vanish.

I am quite confident that the answer to the first question is no (for this exact version):

I think you need to be careful the relativity of your assumptions at least in the first case. Consider two sequences of functions $\hat{f}_n$ with $|\hat{f}_n(x)| < C$ and $\hat{g}_n$ with $c < |\hat{g}_n(x)| < C$ for all $n \in \mathbb{N}$ and $x \in \mathbb{R}^d$.

Now define $f_n = \frac{1}{n^2} \hat{f}_n$ and $g_n = \frac{1}{n} \hat{g}_n$. For sure, $f_n, g_n \rightarrow 0$ and $$|\frac{f_n}{g_n}| < \frac{C}{nc} \rightarrow 0.$$ At the same time, this rescaling does not change minimizers, so $f_n/g_n$ has the same as $\hat{f}_n/\hat{g}_n$. The only condition however that remains with the hat functions is boundedness.

Now you can surely construct two sequences of hat functions that are bounded, but for which the influence of $g$ does not vanish.

I think you need to be careful the relativity of your assumptions at least in the first case. Consider two sequences of functions $\hat{f}_n$ with $|\hat{f}_n(x)| < C$ and $\hat{g}_n(x)$ with $c < |\hat{g}_n| < C$ for all $n \in \mathbb{N}$ and $x \in \mathbb{R}^d$.

Now define $f_n = \frac{1}{n^2} \hat{f}_n$ and $g_n = \frac{1}{n} \hat{g}_n$. For sure, $f_n, g_n \rightarrow 0$ and $$|\frac{f_n}{g_n}| < \frac{C}{nc} \rightarrow 0.$$ At the same time, this rescaling does not change minimizers, so $f_n/g_n$ has the same as $\hat{f}_n/\hat{g}_n$. The only condition however that remains with the hat functions is boundedness.

Then take for example $\hat{f} \equiv 1$ and $\hat{g}(x) = 2 - |x - \frac{1}{n}|$ for $|x| < 1$, $\hat{g}_n(x) = 1 - \frac{1}{n}$, for $x \leq 1$ and $\hat{g}_n(x) = 1 + \frac{1}{n}$ otherwise. For sure the influence of $g_n$ does thus not vanish.

I am quite confident that the answer to the first question is no (for this exact version):

I think you need to be careful the relativity of your assumptions at least in the first case. Consider two sequences of functions $\hat{f}_n$ with $|\hat{f}_n(x)| < C$ and $\hat{g}_n(x)$ with $c < |\hat{g}_n| < C$ for all $n \in \mathbb{N}$ and $x \in \mathbb{R}^d$.

Now define $f_n = \frac{1}{n^2} \hat{f}_n$ and $g_n = \frac{1}{n} \hat{g}_n$. For sure, $f_n, g_n \rightarrow 0$ and $$|\frac{f_n}{g_n}| < \frac{C}{nc} \rightarrow 0.$$ At the same time, this rescaling does not change minimizers, so $f_n/g_n$ has the same as $\hat{f}_n/\hat{g}_n$. The only condition however that remains with the hat functions is boundedness.

Now you can surely construct two sequences of hat functions that are bounded, but for which the influence of $g$ does not vanish.

I think you need to be careful the relativity of your assumptions at least in the first case. Consider two sequences of functions $\hat{f}_n$ with $|\hat{f}_n(x)| < C$ and $\hat{g}_n(x)$ with $c < |\hat{g}_n| < C$ for all $n \in \mathbb{N}$ and $x \in \mathbb{R}^d$.

Now define $f_n = \frac{1}{n^2} \hat{f}_n$ and $g_n = \frac{1}{n} \hat{g}_n$. For sure, $f_n, g_n \rightarrow 0$ and $$|\frac{f_n}{g_n}| < \frac{C}{nc} \rightarrow 0.$$ At the same time, this rescaling does not change minimizers, so $f_n/g_n$ has the same as $\hat{f}_n/\hat{g}_n$. The only condition however that remains with the hat functions is boundedness.

Then take for example $\hat{f} = 1$ and $\hat{g} = 1 + |x - \frac{1}{n}|$. For sure the influence of $g_n$ does thus not vanish.

I think you need to be careful the relativity of your assumptions at least in the first case. Consider two sequences of functions $\hat{f}_n$ with $|\hat{f}_n(x)| < C$ and $\hat{g}_n(x)$ with $c < |\hat{g}_n| < C$ for all $n \in \mathbb{N}$ and $x \in \mathbb{R}^d$.

Now define $f_n = \frac{1}{n^2} \hat{f}_n$ and $g_n = \frac{1}{n} \hat{g}_n$. For sure, $f_n, g_n \rightarrow 0$ and $$|\frac{f_n}{g_n}| < \frac{C}{nc} \rightarrow 0.$$ At the same time, this rescaling does not change minimizers, so $f_n/g_n$ has the same as $\hat{f}_n/\hat{g}_n$. The only condition however that remains with the hat functions is boundedness.

Then take for example $\hat{f} \equiv 1$ and $\hat{g}(x) = 2 - |x - \frac{1}{n}|$ for $|x| < 1$, $\hat{g}_n(x) = 1 - \frac{1}{n}$, for $x \leq 1$ and $\hat{g}_n(x) = 1 + \frac{1}{n}$ otherwise. For sure the influence of $g_n$ does thus not vanish.