Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

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1answer
67 views

$C^2$ Smoothing of absolute value

I am looking for a function $f\colon \mathbb{R}\to\mathbb{R}$ such that $$g(x)=\begin{cases} f(x), &\text{if }| x|<k \\ | x|, &\text{otherwise}\end{cases}$$ is $C^2$ or $C^\infty$ (at ...
1
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1answer
146 views

How to find a surface from two lines?

sorry if this is a basic problem but I don't know where to start looking. Imagine two perpendicular lines ("profiles") in a "$T$" spatial arrangement. The lines are arbitrary (empirical functions ...
6
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0answers
433 views

What is the maximum overshoot of interpolating splines in $d$ dimensions?

Consider cubic splines $s( x, y )$ which interpolate values $y = \{ y_0, y_1, \dots,y_n \}$, on the uniform grid $\{ 0, 1,\dots, n \}$. Fix $s''(0) = s''(n) = 0$ (natural splines). How big can ...
1
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1answer
149 views

Standard Interpolation between Bochner spaces

I've read the following in a few papers: Given: Let $\Omega \subset R^d$ be a Lipschitz domain. A sequence $f_n$ converges strongly to $f$ in $L^2(0,T;L^2(\Omega))$ and weakly in ...
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0answers
73 views

How do you interpolate the local maxima of a set of points in more than 3 dimensions?

I have a set of about 400 points each with 6 coordinates and one scalar value. How can I find out where the local maxima are?
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1answer
324 views

Calculating Log-likelihood using Raphson and Jacobian matrices?

I am reading the following paper: http://www.ntuzov.com/Nik_Site/Niks_files/Research/papers/stat_arb/Ahmed_2009.pdf and in particular page 13. I want to try and calculate lambda_t(p) = exp^(Beta^T ...
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1answer
551 views

Spline Interpolation

I have four questions about splines. Any help would be greatly appreciated. 1) Boundary conditions for cubic spline interpolation to a set of data $a=x{}_{1}<x2<...<x_{m} , $ like for ...
3
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1answer
1k views

Is this algorithm an example of exponential interpolation?

We have an algorithm for calibrating an O2 (oxygen) sensor I'm trying to get my head around. The original author is gone and away and the whole thing seems to generally work, but I'd like to verify ...
1
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0answers
29 views

Interpolation of linear operators

If $T$ is a bounded linear operator from $L^{p_1}$ into a homogeneous Lipschitz space of order, say $\lambda.$ Further if $T$ is also bounded from $L^{p_2}$ into $L^{q}$ for some $p_1,p_2,$ and ...
1
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1answer
48 views

How to fit a formula to three data points?

I need a very basic formula that will be used to determine a CSS line-height based on a provided font pixel size. So in essence, I need the formula to covert ...
2
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1answer
288 views

How do I apply the Fast Multipole Method to Thin Plate Spline interpolation?

I have n scattered measurements of elevation, z, as a function of x and y coordinate: $ \{(x_i,y_i,z_i)\}_{i=1..n}$ that I want to interpolate so that I find z(x,y) for all x and y. Using Thin Plate ...
3
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2answers
146 views

How to fit a function that depends on several nominal and one real variable?

I have data that map several nominal variables and one real parameter into a real value. For example: ...
1
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1answer
377 views

Interpolating missing points in 3D data-set

Given the following x,y,z points (z is actually a signal strength indicator in dBm): ...
3
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1answer
156 views

How can I add a curve to my otherwise linear values?

I have built an interactive map for the Web that transitions smoothly from lon/lat point to lon/lat point. The duration of the transition is calculated dynamically and depends on the distance between ...
2
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0answers
62 views

Smooth detour from one function to another

Suppose I am given two smooth functions $f$ and $g$ on the real line and real numbers $a<b$ such that $f(a)<g(b)$ and $f'(a),g'(b)\ge0$ I want to get a smooth $H:\mathbb R\rightarrow\mathbb R$ ...
1
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2answers
54 views

Interpolation in 2-D co-ordinate system

Suppose there is a function f defined on (x, y) such that x, y $\in$ (-$\infty$, $\infty$), but function is not known. Let n data points are given f($X_0$, $Y_0$) = $Z_0$ f($X_1$, $Y_1$) = $Z_1$ ...
5
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3answers
313 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
1
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1answer
2k views

Minimum surface Curvature Interpolation Method

In this paper about Interpolation Methods, I am trying to learn Minimum curvature method. I have not done partial differential equations before; hence I am finding it tough to penetrate through this ...
1
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2answers
201 views

What's the best way to calculate all of the points for a curve given only a few points?

I've been reading up on curves, polynomials, splines, knots, etc., and I could definitely use some help. (I'm writing open source code, if that makes a difference.) Given two end points and any ...
2
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0answers
22 views

Analogue of Helly’s theorem for non-exact interpolation

Let $\overrightarrow{x}=(x_1,x_2, \ldots ,x_n),\overrightarrow{a}=(a_1,a_2, \ldots ,a_n)$ and $\overrightarrow{b}=(b_1,b_2, \ldots ,b_n)$ be vectors in ${\mathbb R}^n$, with $a_k \leq b_k$ for every ...
2
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3answers
511 views

The Weierstrass Approximation Theorem Vs The Runge's Phenomenon

I am learning about different interpolation methods in my internship. Today as I was looking this article on Wikipedia to learn about the Runge's Phenomenon exhibited by Polynomial Interpolation. I ...
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0answers
52 views

interpolation and Vandermonde

Looking at a problem of interpolation, I find a Vandermonde type matrix. To be precise I consider the following, let $$A(z)= \sum_{i=1}^p \sum_{j=2}^{n_i+1}\frac{a_{i}^j}{(z-z_i)^j}$$ where the $z_i$ ...
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0answers
205 views

Best method of interpolation?

I am learning different interpolation methods, and their pros and cons. Which interpolation method do you think is the best for practical use? If you can give me links to research papers about various ...
2
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2answers
64 views

Converting data to a specified range

I am trying to convert data to a particular range of 0-10. Actual Data may vary from - 50000 - 26214400. I have broken this down in to 4 parts as follows - ...
1
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0answers
98 views

Quaternion barycentric interpolation

Let's say that i have a set of quaternions, each representing a 3-angle orientation. And with each quaternion is associated a real value (let's say a speed value for explanation's sake). Now with an ...
3
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0answers
118 views

Interpolation method

I am researching about different interpolation methods, and their pros and cons. Please help me to understand ideas behind Gaussian interpolation method. Although, I looked at different papers for ...
4
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1answer
93 views

Interpolation between iterated logarithms

I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$ Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for ...
2
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1answer
842 views

Thin Plate Spline interpolation of scattered $z(x,y)$ data

I am trying to understand Thin Plate Spline interpolation of scattered data. As I understand it TPS is just a special case of Radial Basis Function interpolation: $$ z(x,y) = p(x,y) + \sum_i ...
2
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0answers
192 views

monotonic smoothing fit to be implemented (in python or other language)

In a post that already exists, implementation-of-monotone-cubic-interpolation, there is a good method for fitting data which necessarily includes all of the given points. But, what if I need to ...
2
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4answers
154 views

Function generation by input $y$ and $x$ values

I wonder if there are such tools, that can output function formulas that match input conditions. Lets say I will make input like that: $y=0, x=0$ $y=1, x=1$ $y=2, x=4$ and tool should ...
0
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2answers
259 views

Why are there problems when interpolating $f(x)=\arctan(10x)$?

Given $f(x)=\arctan(10x)$, there would be a problem when we interpolate it by using Lagrange's method. This would have something to do with the derivatives of $f(x)$. I plotted some derivatives of $f$ ...
2
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0answers
110 views

Minimal surface representation from a 3D contour

I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane and I want to obtain a 3D triangular mesh representation of the surface inside ...
2
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0answers
157 views

Ellipse radius interpolation with different radiuses

I am writing a library for graphical LCDs and I want to incorporate a function to draw a circle on the screen. I have already succeeded in drawing simple circles, however, I want to be able to pass a ...
0
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1answer
453 views

Coefficients of Newton interpolation polynomial

Given distinct $y_0,...,y_m$ in $\mathbb R$, let $N_m(x)$ be the Newton interpolation polynomial of degree $m$. That is, $N_m(x) = \sum_{n=0}^{m}a_nw_n(x)$ where $w_0 = 1$, $w_n(x) = ...
3
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1answer
293 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
0
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2answers
1k views

Forcing Bezier Interpolation

I found this very informative site that discusses forcing bezier interpolation and the site gives formulae for calculating the control points so that the curve goes through a set of four points, y0, ...
5
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4answers
581 views

Proof of Schur's test via Young's inequality

I am able to prove the following generalization of Schur's test using the Riesz-Thorin interpolation theorem, however I have been stuck for days now trying to prove it using Young's inequality: Let ...
1
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0answers
426 views

Multigrid Interpolation and Restriction operators

I have a question about the restriction and the interpolation operators of a Multigrid algorithm. Let those be given: The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ ...
6
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0answers
389 views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
1
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0answers
233 views

Non-linear interpolation. (1D Perlin Noise)

In this document (http://webstaff.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf) about Perlin (and Simplex) Noise you can find an explanation about 1D Perlin Noise (at the top). ...
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0answers
220 views

Interpolate to 3D functions

Hy, In math classes, I've learned that given some points in 2D space: a(1,2), b(7,3), c(8,5),... You can find an equation that goes through these points (using interpolation). Now I was wondering if ...
2
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1answer
71 views

Interpolation with degree restriction

(using $f[x_1, ... , x_n]$ to denote the forward difference operator) I have a polynomial $P(x)$ interpolating $5$ points $x_0, ... , x_4$ and $2$ derivative values $x_0, x_3$ across an evenly spaced ...
1
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1answer
73 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
2
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1answer
148 views

Nontrivial expansion of a multivariate power series in form of a single variable series?

I am trying to interpolate a function defined over a three-dimensional real space: $$f: R^3\rightarrow R\\(x,y,z)\rightarrow f(x,y,z)$$ Let assume I have $N_1 N_2 N_3$ points in the space which form ...
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0answers
29 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
1
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2answers
1k views

Easing function, constant velocity then decelerate to zero

I'm trying to write an interpolator for a translate animation, and I'm stuck. The animation passes a single value to the function. This value maps a value representing the elapsed fraction of an ...
2
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1answer
57 views

Is there a nice way to interpret this matrix equation that comes up in the context of least squares

So I am working on this problem with fitting a second degree polynomial of the form $y=a_1x^2+a_2x+a_3$ to four points using least squares. One of the parts of the problem is to write out the matrix ...
0
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2answers
130 views

Finding t in basic geometric interpolation

I have the following problem related to interpolation and was wondering if anyone had any ideas? In the following diagram, there exists known-points: A, B, C, D, E, unknown points F, G. I'm trying ...
3
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1answer
108 views

Linear, Bi-linear or better

I have been writing some code to do some interpolation of 2D data on an irregular grid. So far what I have done is: Triangulate the known points using Delaunay. Find the vertices of the triangles ...
2
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1answer
1k views

How does one interpolate between polar coordinates?

I'm finding that when I try to use the standard methods of interpolation in polar space, the result is not what I would expect. For example, when interpolating between the following polar coordinates: ...