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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144 views

estimating a particular analytic function on a bounded sector.

Let $f(z)$ be an analytic function on $C^+=\{\Re z>0\}$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
6
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430 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 ...
6
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387 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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0answers
123 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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1k 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 ...
3
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39 views

Lagrangian interpolation

I have a high-school student doing a modeling project using interpolation. (Hopefully someone understands what I mean without having to write out explicit examples, as I only have screenshots.) When ...
3
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0answers
204 views

How does 2D kriging interpolation work?

I have a grid of points Example ...
3
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88 views

Random multipliers of polynomial values at known points in $\mathbb{Z}_p$

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
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117 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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267 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 ...
3
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0answers
302 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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72 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
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44 views

From a set of vertices, find smallest polytope enclosing another point

Out of a set of vertices $V=\{\vec v_i\in \mathbb R^D\}$, I am constructing a piecewise linear interpolating function $f:\mathbf{conv}(V)\rightarrow R$ as follows: given a point $\vec d\in ...
2
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0answers
94 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
2
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108 views

Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
2
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0answers
38 views

Regarding the Lebesgue constant for interpolation

I have a question regarding Lebesgue constant $\Lambda_{n}\left(\boldsymbol{\chi}\right)$, with which the worst case error between an interpolant $p\left(\boldsymbol{x}\right)$ and the function which ...
2
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0answers
52 views

Formula for $s_n = \sum_{i = 1}^n i^3$ Newton's Forward Difference Interpolation

Use Newton's Forward Difference formula to find an expression for $$ S_n = \sum_{i = 1}^{n} i^3$$ This is from an Introductory Numerical Analysis paper. I cannot figure out the connection ...
2
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35 views

Convex interpolation between two points with given derivatives

Let's say I have two real values $x_1$ and $x_2$, to each of which I associate $y_i$ and $y'_i$ satisfying $$ (y_2-y_1)(y'_2 - y'_1) \geq 0. \tag{1} $$ I would like to find a polynomial ...
2
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44 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
35 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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73 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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0answers
143 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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0answers
605 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 ...
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50 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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0answers
403 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 ...
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 $H:\mathbb R\rightarrow\mathbb R$ ...
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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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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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
155 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
199 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
95 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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0answers
630 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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0answers
435 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 ...
2
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0answers
114 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 ...
2
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0answers
170 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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0answers
404 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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0answers
47 views

Numerical mathematics, Lagrange interpolation

I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical ...
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0answers
31 views

Interpolation at “extreme values.”

I am working with a meteorologist on a project. We are pulling data from METAR Observation stations on several variables (such as temperature, dew point, wind speed, etc.) throughout time. ...
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0answers
24 views

How to select the number of nodes in a spline interpolation?

I am writing a program to test the precision of different methods for imputing missing data in a time series. One of the methods I am going to test is a natural cubic spline interpolation. I'll be ...
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0answers
16 views

Interpolation of jacobians for point wise defined transformation

Here is my question: let's say that I have a transformation function T from the image A to the image B which is pointwise defined. That is, T(x) = J, where J is the jacobian representing the ...
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0answers
38 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
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0answers
17 views

$(x,y)$ points extrapolation

My process is generating $x,y$ points. The points are result of tracking movement of an object so its a bit inertial. The tracked object can't change direction rapidly. I need to extrapolate data ...
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0answers
54 views

How can I interpolate the following data and get the equation?

I want to interpolate between three vectors which have 8 rows in each. For eg. X = [1 2 3 4 5 6 7 8], Y = [7 8 9 10 11 12 13 14 ], Z = [15 16 17 18 19 20 21 22] I want to find the equation Z = ...
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0answers
43 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
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0answers
48 views

Polynomial Evaluation

My question is to some extent related to cryptography, but I'd like the mathematicians answer my question, please (as their answers are usualy more clearer than cryptographers). Consider I have a ...
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0answers
29 views

Do nth degree polynomials derived using Least Squares Interpolation always have n+1 intersections with the function?

I have recently studied Interpolation Techniques in my College Numerical Methods class and I have this question: If we have a function $f(x)$ and we are asked to use Least Squares Interpolation(LSI) ...
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0answers
33 views

Creating FEM mesh for image region — what is the most suitable shape function?

I wish to create a FEM mesh to solve an inverse elasticity problem, for an irregular domain. This domain is given by a medical image, so it is discretised and each square on the grid has one scalar ...
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0answers
24 views

Sum of $p$th powers using polynomial interpolation

It is well known that the sum of the first $n$ $p$th-powers is polynomial in $n$ and is given by: $$ \sum_{k=1}^n k^p = \frac{1}{p+1} \sum_{j=0}^p (-1)^j {p+1 \choose j} B_j n^{p+1-j} $$ where $B_i$ ...
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20 views

Interpolation of the metric tensor

I am currently facing the following problem. I have a Riemannian manifold, where the metric is only known at certain points. Are there some standard strategy to interpolate the metric in other points ...