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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5
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220 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 ...
4
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115 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
4
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0answers
256 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 ...
3
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108 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 ...
3
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900 views

Computation of coefficients of Lagrange polynomials

For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the ...
2
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35 views

Interpolation of iterated logarithms

$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$ gives an interpolation between ...
2
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0answers
30 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
2
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47 views

Lagrange's interpolation to solve for 0 of y(x)

I have the data composing of 7 elements x is from 0 → 3 incrementing by 0.5 y is from 1.8241 → -1.5427 I am supposed to use Lagrange's interpolation of three nearest neighbor data points. I am ...
2
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90 views

lagrange interpolation question here

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0, x_1=0.25 , x_2=0.5, x_3=1$ a)Find Lagrange polynomial for this function. c)Find the real approximation error. d)Find the limit of the ...
2
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223 views

Interpolation between curves

I am trying to do a 2D interpolation between two curves as you can see from the attached picture. where $a$ is a varrying paramter. $f_{a_1}$ and $f_{a_2}$ are known, or at least I can perform ...
2
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38 views

Exercise 1.3.3(c) of GTM249(Classical Fourier Analysis)

Exercise 1.3.3(c) Let $0<p_0<p<p_1<\infty$ and let $T$ be an operator as in Theorem 1.3.2($\|T(f)\|_{L^{p_0,\infty}(Y)}\leq A_0\|f\|_{L^{p_0}(X)}$ for all $f\in L^{p_0}(X)$ and ...
2
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54 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 $H:\mathbb R\rightarrow\mathbb R$ ...
2
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0answers
20 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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0answers
108 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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74 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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102 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 ...
2
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0answers
155 views

B-Spline Definition

I'm currently working on my master's project. For this, I rely on one PhD-thesis in which I found a statement I do not understand. Unfortunately, the author hasn't answered to my mails yet, so I have ...
2
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0answers
92 views

variogramas kriging

In order to find the altitude of a surface, we use the following method. My question is: What is the name of this method? We have $$ A=\begin{pmatrix} a_{1,1} & .. & .. & ...
2
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432 views

Interpolating polynomial with Chebyshev nodes

I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points. ...
2
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269 views

Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
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184 views

Runge function error second factor

I'm currently learning about the Runge function. On Wikipedia, I read the following: Consider the function: $ \dfrac{1}{1+25x^2}$ Runge found that if this function is interpolated at ...
2
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0answers
223 views

linear interpolation error estimate for non-smooth function

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
2
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98 views

explicit error bounds for Multivariate interpolation

I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
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139 views

(Experimental) Can it be shown that this extension of the secant-interpolation has quadratic convergence?

Background: I needed some efficient but simple interpolation-methods aside of Newton's iteration because I want to have it in contexts, where the derivative of a function is not always known. So an ...
2
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224 views

Is there software that interpolates/extrapolates data using a discrete Fourier?

I've read various methods of Fourier interpolation and extrapolation detailed in articles such as Interpolation and Extrapolation Using a High-Resolution Discrete Fourier Transform—so what I'm ...
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22 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
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23 views

A problem on interpolation

Given $X(0) \in \Bbb R$ and a continuous and bounded real-valued function $g(.)$ from $\Bbb R \to \Bbb R$, for $\delta > 0$ define the sequence $\{X_n^\delta\}$ by $\{X_0^\delta\} = X(0)$ and $$ ...
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19 views

Newton Interpolation jump in notes

I am told, The general expression for a second order polynomial that passes through 3 points $(x_1,y_1),(x_2,y_2)$, and $(x_3,y_3)$ can be written as: $$ p_2(x)=b_0+b_1x+b_2x^2 \tag1$$ which I am ...
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0answers
100 views

Lagrange interpolating polynomials question?

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0$, $x_1=0.25$ , $x_2=0.5$, $x_3=1$ a)Find Lagrange polynomial for this function. So $L_3(x)=f_0(x) l_0(x)+f_1(x) l_1(x)+f_2(x) ...
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0answers
18 views

Differntiability properties of kriging interpolation

What are the differentiability properties of Kriging interpolation functions? Specifically, I am interested in using it to create a random realization of a 1D function using a regularly spaced grid of ...
1
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0answers
59 views

Differentiation by interpolation.

I am asked to show that the formula: $$ f'(x)\sim \sum_{i=0}^n A_i f(x_i) $$ which is derived from differentiating the interpolation polynomial is similar to that derived from checking/evaluating the ...
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23 views

$M(x)$ and $L(x)$ interpolate $f(x)$ on $n+1$ points. Show, that $f(x)$ lies between $L(x)$ and $M(x)$

We have $n+2$ points $x_0 \lt x_1 \lt x_2 ... \lt x_{n+1}$. We have two polynomials - $L$ and $M$. $L(x)$ interpolates $f(x)$ on points $x_0,...,x_n$ and $M(x)$ does so on $x_1,...,x_{n+1}$. The ...
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0answers
44 views

How stable is my quaternion interpolation?

after some experimentation to optimize slerp I found that finding the middle between quaternion is rather cheap (for $t=0.5$) in particular: (with $\theta$ the angle between $q_1$ and $q_1$) ...
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0answers
72 views

Successive parabolic interpolator for sub pixel interpolation

Is it possible to use successive parabolic interpolator for doing sub pixel interpolation. In the case of non sub pixel interpolation it is very easy to apply successive parabolic interpolation as ...
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0answers
488 views

How to calculate a frequency curve from a sample of values, and interpolate between curves

I have a set of about 500 values, from which I'm currently plotting a histogram. I'd like to plot a frequency curve, i.e. go from the left two graphs to the rightmost on the below image borrowed from ...
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0answers
42 views

Given a finite set of points construct a polynomial that meets the points.

Say I have a set of points in $\mathbb{Z}^3 \times \mathbb{Z}_2$ each of which represent part of a mapping $(z_1, z_2, z_3) \mapsto z_4 \in \mathbb{Z}_2$. How do I find the the simplest polynomial ...
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0answers
216 views

Shannon vs dirichlet kernel interpolation method for signal reconstruction

I am currently studying fourier transform, and especially the way that the signal could be reconstructed from its spectrum. In many lectures, I have seen the shannon interpolation method to ...
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0answers
68 views

What do quadratic smoothing splines minimize?

Cubic smoothing splines minimize a combination of Interpolation cost and Smoothness (roughness) cost: $\qquad$ min Icost + $\lambda$ Scost where $\qquad$ Icost $\equiv \sum (Y_i - \mu(x_i))^2$ ...
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0answers
62 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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0answers
27 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 ...
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0answers
173 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 ...
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0answers
77 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 ...
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0answers
200 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[ ...
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0answers
163 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
157 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 ...
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0answers
22 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 ...
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0answers
75 views

What are the constants in the relationship between point density and median point distance?

Given $\rho$ particles uniformly distributed on a plane within a unit square ($\rho > 1$), each particle has another particle that is closest to it; the median of those nearest distances is called ...
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60 views

Analogue for finite sums of $$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) (*)$$

Please help me to find an analogue for finite sums of $$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) (*)$$ where $ \bar{f}(t)=\frac{1}{t}\int_{a}^{t} ...
1
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0answers
41 views

Interpolation Inequality in finite dimensional linear space

How can we prove - $$|\dot{u}(x) - \tilde{u_h}(x)| \leqslant h \left[\max_{0 \leqslant y\leqslant1}\ddot{u}(y)\right],$$ where $0\leqslant x\leqslant 1$ and $\tilde{u_h}$ is interpolant of $u$.
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0answers
455 views

Trapezoidal Rule for Numerical Integration

If the trapezoidal rule approximates an integral with trapezoids, then I thought (and was tought in high school) that the formula is: $ \frac{h}{2}(f(x) + f(x + h))$ Where $h$ is the distance between ...