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:
- 7,325
6
votes
Accepted
Sign of 3j symbol (in view of interpolation)
Wolfram contains the following formula which should make your calculations easy:
If $\ell_1+\ell_2+\ell_3=2g$ then $$\begin{pmatrix}
\ell_1 &\ell_2 &\ell_3\\
0&0&0
\end{pmatrix}=$$
...
- 83.4k
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
$$
\det \begin{pmatrix} f(a) & g(a) \\ f(b) & g(b) \...
- 18.4k
6
votes
Accepted
Wasserstein interpolation between two probability measures on a metric space
The following discussion is based on the book Gradient Flows by Ambrosio, Gigli, and Savare (2008).
Consider $p$-Wasserstein distance with $p>1$ on a Hilbert space (for the sake of uniqueness). ...
- 412
6
votes
Accepted
Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise
The result that you mention in the first part of your question is a classical result by Faber
G. Faber, Uber die interpolatorsche Darstellung stetiger Funktionen, Jahresber. der deutschen Math. ...
- 3,596
6
votes
Accepted
For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve
Let $n \ge 2$. Given any points $x_0 < x_1 < \dots < x_n$, there is a quadratic function positive at all those points, but negative somewhere in $[x_0,x_n]$.
Indeed, let $c \in [x_0,x_n]$...
- 39.4k
6
votes
Accepted
Interpolation of product spaces
Yes, interpolation on product spaces works componentwise, so $$\Bigl(\prod_{i=1}^n X_i,\prod_{i=1}^n Y_i\Bigr) = \prod_{i=1}^n (X_i,Y_i)$$ for any interpolation functor $(\cdot,\cdot)$ even with equal ...
- 1,883
5
votes
Accepted
Elegant / Canonical way to Extend Integer Iterates of a Function to a Real Parameter
Notice that $e^x$ does not have an inverse on the whole real line.
Extension of iterates is possible if $f$ has a fixed point $x_0$. Suppose for example, that this fixed point is repelling that is $f(...
5
votes
Asymptotic behavior of sum linked with Lagrange interpolation
For a finite non-empty set $\Omega=\{\omega_1,\dots,\omega_{N+1}\}\subset \mathbb{R}$, denote by $\Phi_{\Omega}$ the linear functional which maps a function $g:\Omega\mapsto \mathbb{R}$ to $$\Phi_{\...
- 94.7k
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 $\...
- 7,773
4
votes
Accepted
Relation between Chebyshev Interpolation and Expansion
There is an excellent explanation of this in Chapter 4 of L. N. Trefethen's Approximation Theory and Approximation Practice (henceforth ATAP; the first 6 chapters are available for free online). I ...
- 1,213
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 ...
- 646
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 ...
- 5,582
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 ...
- 3,596
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 ...
- 1,213
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
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) ...
- 646
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 $(\infty,1)$ ...
- 24.5k
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
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 ...
- 1,189
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 ...
- 3,596
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: ...
- 33.9k
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 $(N-2) \times (N-2)$ ...
- 646
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 ...
- 52.6k
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 ...
- 4,214
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, ...
- 9,159
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 ...
- 1,307
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-...
2
votes
Accepted
Interpolation between weighted $L^p$ spaces
I think what you need is in the following paper:
E. M. Stein and G. Weiss, Interpolation of operators with change of measures,
Transactions of the American Mathematical Society, Vol. 87 (1958), pp. ...
- 12.7k
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