Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum points of $s$? This bound may depend on $n$ and/or the number of local extremum points of $f$. If not, are there stronger conditions by which this is true?

An easy condition on $f$ — If $f\in C^4$, then $|f'-s'|<C h^3$, where $h=\max\limits_{j=1,\ldots n} |t_j - t_{j-1}|$. So, if $|f'|>\alpha >0$ on $[a,b]$, then for sufficiently small $h$ we have that $|s'|>\frac{\alpha}{2} > 0$. So, to sharpen the original question: Can anything be said about such situations where $f'=0$ on finitely many points in $[a,b]$?

Intuition: The natural cubic spline minimizes the curvature $\|s''\|_2$, and so it seems that the arc-length of the graph of $s$ is small. Therefore, it might imply that $s$ doesn't oscillate "too much", i.e., no more then the function $f$ it interpolates. However, I could not find or prove anything of that sort myself, and a counter-example may as well exist.

Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum points of $s$? This bound may depend on $n$ and/or the number of local extremum points of $f$. If not, are there stronger conditions by which this is true?

An easy condition on $f$ — If $f\in C^4$, then $|f'-s'|<C h^3$, where $h=\max\limits_{j=1,\ldots n} |t_j - t_{j-1}|$. So, if $|f'|>\alpha >0$ on $[a,b]$, then for sufficiently small $h$ we have that $|s'|>\frac{\alpha}{2} > 0$. So, to sharpen the original question: Can anything be said about such situations where $f'=0$ on finitely many points in $[a,b]$?

Intuition: The natural cubic spline minimizes the curvature $\|s''\|_2$, and so it seems that the arc-length of the graph of $s$ is small. Therefore, it might imply that $s$ doesn't oscillate "too much", i.e., no more than the function $f$ it interpolates. However, I could not find or prove anything of that sort myself, and a counter-example may as well exist.

Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum point of $s$? This bound may depend on $n$ and/or the number of local extremum points of $f$. If not, are there stronger coditions by which this is true?

An easy condition on $f$ - If $f\in C^4$, then $|f'-s'|<C h^3$, where $h=\max\limits_{j=1,\ldots n} |t_j - t_{j-1}|$. So, if $|f'|>\alpha >0$ on $[a,b]$, then for sufficiently small $h$ we have that $|s'|>\frac{\alpha}{2} > 0$. So, to Sharpen the original question: Can anything be said about such situations where $f'0$ on finitely many points in $[a,b]$?

Intuition: The natural cubic spline minimizes the curvature $\|s''\|_2$, and so it seems that the arc-length of the graph of $s$ is small. Therefore, it might imply that $s$ doesn't oscillate "too much", i.e., more then the function it interpolates, $f$. However, I could not find or prove anything of that sort myself, and a counter-example may as well exist.

Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum points of $s$? This bound may depend on $n$ and/or the number of local extremum points of $f$. If not, are there stronger conditions by which this is true?

An easy condition on $f$ — If $f\in C^4$, then $|f'-s'|<C h^3$, where $h=\max\limits_{j=1,\ldots n} |t_j - t_{j-1}|$. So, if $|f'|>\alpha >0$ on $[a,b]$, then for sufficiently small $h$ we have that $|s'|>\frac{\alpha}{2} > 0$. So, to sharpen the original question: Can anything be said about such situations where $f'=0$ on finitely many points in $[a,b]$?

Intuition: The natural cubic spline minimizes the curvature $\|s''\|_2$, and so it seems that the arc-length of the graph of $s$ is small. Therefore, it might imply that $s$ doesn't oscillate "too much", i.e., no more then the function $f$ it interpolates. However, I could not find or prove anything of that sort myself, and a counter-example may as well exist.

Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum point of $s$? This bound may depend on $n$ and/or the number of local extremum points of $f$. If not, are there stronger coditions by which this is true?

Intuition: The natural cubic spline minimizes the curvature $\|s''\|_2$, and so it seems that the arc-length of the graph of $s$ is small. Therefore, it might imply that $s$ doesn't oscillate "too much", i.e., more then the function it interpolates, $f$. However, I could not find or prove anything of that sort myself, and a counter-example may as well exist.

Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum point of $s$? This bound may depend on $n$ and/or the number of local extremum points of $f$. If not, are there stronger coditions by which this is true?

An easy condition on $f$ - If $f\in C^4$, then $|f'-s'|<C h^3$, where $h=\max\limits_{j=1,\ldots n} |t_j - t_{j-1}|$. So, if $|f'|>\alpha >0$ on $[a,b]$, then for sufficiently small $h$ we have that $|s'|>\frac{\alpha}{2} > 0$. So, to Sharpen the original question: Can anything be said about such situations where $f'0$ on finitely many points in $[a,b]$?

Intuition: The natural cubic spline minimizes the curvature $\|s''\|_2$, and so it seems that the arc-length of the graph of $s$ is small. Therefore, it might imply that $s$ doesn't oscillate "too much", i.e., more then the function it interpolates, $f$. However, I could not find or prove anything of that sort myself, and a counter-example may as well exist.