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10 votes
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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:
Dustin G. Mixon's user avatar
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) \...
Christian Remling's user avatar
6 votes
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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. ...
user111's user avatar
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6 votes
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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). ...
O. Richard's user avatar
6 votes
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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 $c \in [x_0,x_n]$...
Gerald Edgar's user avatar
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6 votes
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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 ...
Hannes's user avatar
  • 2,265
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 $A$ ...
Matt Werenski's user avatar
6 votes
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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}...
Iosif Pinelis's user avatar
5 votes
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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(...
Alexandre Eremenko's user avatar
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_{\...
Fedor Petrov's user avatar
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 ...
Iddo Hanniel's user avatar
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 $\...
Henry.L's user avatar
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4 votes
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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 ...
Peter Taylor's user avatar
  • 6,786
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 ...
Iddo Hanniel's user avatar
3 votes
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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) ...
Iddo Hanniel's user avatar
3 votes
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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 ...
Mateusz Kwaśnicki's user avatar
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 $(\infty,1)$ ...
Piotr Hajlasz's user avatar
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 ...
David Ketcheson's user avatar
3 votes
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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, ...
Ryan Tully-Doyle's user avatar
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 ...
user111's user avatar
  • 3,904
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 ...
user521337's user avatar
  • 1,199
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 ...
user111's user avatar
  • 3,904
3 votes
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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: ...
Willie Wong's user avatar
  • 38.4k
3 votes
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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 $(N-2) \times (N-2)$ ...
Iddo Hanniel's user avatar
3 votes
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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 ...
Robert Israel's user avatar
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 ...
Giorgio Metafune's user avatar
3 votes
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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, ...
Tom Copeland's user avatar
  • 9,975
3 votes
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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 ...
Mark Schultz-Wu's user avatar
3 votes
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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-...
Alexandre Eremenko's user avatar

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