# Tag Info

Accepted

### Do splines preserve monotonicity?

I assume you mean nondecreasing, i.e., monotonically increasing. If so, the answer is "no." For example, interpolate (0,0), (1,10) and (2,11) with a natural cubic spline:
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### Can a polynomial be evaluated from evaluations of partial interpolations? (Or: can the unique solution of congruences be written in a certain way?)

No. The interpolating polynomial is a weighted sum $a(x) = \sum_{x_i} a(x_i) P_i(x)$ and the independence of the $a(x_i)$ from each other imposes the independence of the $P_i$ from each other, which ...
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### Do splines preserve monotonicity?

Not only is the answer "no", but for any number $N$ you can construct a monotone function and sample it such that the natural spline approximation will have $N$ extremum points. See the figure (and ...
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### Interpolation nodes for linear spline (piecewise-linear) interpolation of $x \ln x$

There is an analytical solution to the problem in the following sense: Given a number $N$, the optimal interpolation points $x_0=0, x_1, ..., x_{N-1}=1$ are the roots of an $(N-2) \times (N-2)$ ...
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### Can this function be interpolated with a small power series

Take an entire function $f$ such that $f(0)=1$ and $f(j) = 0$ for all nonzero integers: an example is $f(z) = \sin(\pi z)/(\pi z)$ for $z \ne 0$, $1$ for $z=0$. The Maclaurin series of $f$ satisfies ...
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### An interpolation inequality

Such an inequality is false. Take $n=1$, $\phi$ a standard cut-off function supported in $[0,1]$ and $f=\sum_n c_n \phi (\frac{x-n}{r_n})$. Then $\|f\|_p \approx \sum_n |c_n|^p r_n$, \$\|f'\|_p \approx ...
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### About Newton's forward and backward interpolation

The forward and backward finite differences and the derivative lower the degree of a polynomial by one. This property underlies the construction of series expansions of polynomials and, therefore, ...
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### Fastest Implementation of polynomial interpolation?

Polynomial interpolation can be done via multiplying a Vandermonde matrix (or its inverse) by your coefficient/evaluation vector --- it is a change of basis on the vector space of polynomials of ...
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