10
votes

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:

6
votes

### A Taylor formula for a Vandermonde-like determinant

This isn't working. Let's look at the simplest case, $N=1$ (but the example works in general). Then the identity you are hoping for becomes
$$
\det \begin{pmatrix} f(a) & g(a) \\ f(b) & g(b) \...

6
votes

Accepted

### Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

The result that you mention in the first part of your question is a classical result by Faber
G. Faber, Uber die interpolatorsche Darstellung stetiger Funktionen, Jahresber. der deutschen Math. ...

6
votes

Accepted

### Wasserstein interpolation between two probability measures on a metric space

The following discussion is based on the book Gradient Flows by Ambrosio, Gigli, and Savare (2008).
Consider $p$-Wasserstein distance with $p>1$ on a Hilbert space (for the sake of uniqueness). ...

6
votes

Accepted

### For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

Let $n \ge 2$. Given any points $x_0 < x_1 < \dots < x_n$, there is a quadratic function positive at all those points, but negative somewhere in $[x_0,x_n]$.
Indeed, let $c \in [x_0,x_n]$...

6
votes

Accepted

### Interpolation of product spaces

Yes, interpolation on product spaces works componentwise, so $$\Bigl(\prod_{i=1}^n X_i,\prod_{i=1}^n Y_i\Bigr) = \prod_{i=1}^n (X_i,Y_i)$$ for any interpolation functor $(\cdot,\cdot)$ even with equal ...

6
votes

### Interpolation between two matrices so that $L^p$ norm is controlled

The constraint that $||Ax||_1 = ||x||_1$ is actually a very strong constraint. It actually implies that $A$ is a signed-permutation matrix, that is the matrix $A \in \{-1,0,1\}^{n \times n}$ and $A$ ...

6
votes

Accepted

### Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$

Counterexample: a smoothed version of the function $f$ given by the formula
$$f(x)=(h/2)^{1/2}(1-\max(0,|x/h-1|))\tag{1}\label{1}$$
with $h\downarrow0$.
Indeed, if we had
$$\|f\|_\infty\le C\,\|f\|_{2}...

5
votes

Accepted

### Elegant / Canonical way to Extend Integer Iterates of a Function to a Real Parameter

Notice that $e^x$ does not have an inverse on the whole real line.
Extension of iterates is possible if $f$ has a fixed point $x_0$. Suppose for example, that this fixed point is repelling that is $f(...

5
votes

### Asymptotic behavior of sum linked with Lagrange interpolation

For a finite non-empty set $\Omega=\{\omega_1,\dots,\omega_{N+1}\}\subset \mathbb{R}$, denote by $\Phi_{\Omega}$ the linear functional which maps a function $g:\Omega\mapsto \mathbb{R}$ to $$\Phi_{\...

4
votes

### Maximum of a B-spline

Using the recursive derivative formula (see for example here):
$$N'_{i,p}(t) = \frac{p}{t_{i+p}-t_i} N_{i,p-1}(t) - \frac{p}{t_{i+p+1}-t_{i+1}} N_{i+1,p-1}(t)$$
We get that the maximum is achieved ...

4
votes

### Variational proof for minimum curvature of cubic splines

See pp.87~107 of
Prenter, Paddy M. Splines and variational methods. Courier Corporation, 2008. especially p.100 where "uniqueness theorem" is proved and spline is defined as minimizer to $\...

4
votes

Accepted

### 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 ...

3
votes

### 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 ...

3
votes

Accepted

### Optimal $L^2$ bounds of cubic spline interpolation

From the properties of integrals it is not hard to derive an upper bound on the $L^2$ norm from the upper bound on the $L^\infty$ norm.
Since you have (from Hall & Meyer) a bound
$\left| f(x)-s(x) ...

3
votes

Accepted

### Interpolation space between $L^1\cap L^2$ and $L^1$

As requested, I post my comment as an answer (although this is not a true answer, just a possibly useful reference; feel free to edit it if this approach works out).
In Section 3 of the article ...

3
votes

### Interpolation space between $L^1\cap L^2$ and $L^1$

In the case of a set of finite measure (Bourgain in the quoted paper deals with the case of finite measure (torus)) we have that $\Vert f\Vert_1\leq C\Vert f\Vert_2$ so we actually have $(\infty,1)$ ...

3
votes

### Cubic interpolating spline – number of extremum points

Can we use the number of local extremum points of $f$ to bound the number of local extremum points of $s$?
No. See https://en.m.wikipedia.org/wiki/Monotone_cubic_interpolation for a counterexample ...

3
votes

Accepted

### Who was first to use reproducing kernels in order to try to solve interpolation problems?

You might look at Carleson's 1958 paper "An interpolation problem for bounded analytic functions". A modern treatment is given in Agler and McCarthy's Pick Interpolation and Hilbert Function Spaces, ...

3
votes

### Interpolation by rational functions reference

Here are four references on the subject (the main ones as far as I know) :
Baker, George A.; Graves-Morris, Peter,
Pad\'e approximants.
Second edition. Encyclopedia of Mathematics and its ...

3
votes

### Does every positive continuous function have a non-negative interpolating polynomial of every degree?

I think I can prove the following : For every continuous function $f:[a,b]\to (0,\infty)$, $\exists n_0>1$ such that for every $n\ge n_0$, there are $n+1$ distinct points in $[a,b]$ such that the ...

3
votes

### Does every positive continuous function have a non-negative interpolating polynomial of every degree?

This question was recently studied in that paper:
F. Charles, M. Campos-Pinto, B. Després, Algorithms for positive polynomial approximation, hal-01527763,
assuming that the function $f$ is Lipschitz ...

3
votes

Accepted

### Interpolation of a trilinear functional

If you take the bilinear operator $T:(f,g) \mapsto \int K(x,y,z) f(y,z) g(x,z) ~\mathrm{d}z$, your three boundedness statements are equivalent to
$T: L^2 \times L^4 \to L^{4/3}$ with norm $C_1$
$T: ...

3
votes

Accepted

### 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 $(N-2) \times (N-2)$ ...

3
votes

Accepted

### 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 ...

3
votes

### 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 ...

3
votes

Accepted

### 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, ...

3
votes

Accepted

### 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 ...

3
votes

Accepted

### Power series whose coefficients are limits of coefficients of polynomial interpolations

In the paper, A note on convergence of Newton interpolating polynomials,
by D. Dimitrov and J. Philipps, Journal of Computational and Applied Mathematics
Volume 51, Issue 1, 30 May 1994, Pages 127-...

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