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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Hardy-Littelwood-Sobolev without Marcinckiewicz?

Here is the statement of the Hardy-Littlewood-Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| ...
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1answer
31 views

Using several curves in 3D to create a surface

I have a set of several closed curves in 3d (like image below is showing my set of curves from 3 views). To clarify my idea, i ask my questions in two different ways showed by diction 1 and diction ...
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2answers
19 views

First Derivative The slope at the sample points

I am trying to implement an interpolation function in C# and one of the parameter is an array of 4 elements, which should contains first derivative of the slope at the sample 4 points. I am not a ...
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0answers
13 views

Estimate accuracy of inaccurate fast function having exact values of slow one

Let’s say we have functions $F$ and $H$ to calculate a series $S$ of integers and that: $S_{i} = H(x_{i}) = F(x_{i}) + e_{i}$ Being $e_{i}$ the error of $F(x_{i})$ to estimate $S_{i}$ The problem ...
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1answer
20 views

Quadratic spline and quadratic interpolation

I am trying to understand what is the difference between quadratic spline and quadratic interpolation. Thank you for any help and advice.
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1answer
22 views

How to calculate the amount of time spent interpolating from one tempo value to another

I am writing a music creation program where the user is allowed to change the tempo throughout the track. If the user had a set tempo or only changed the tempo at discrete intervals I could easily ...
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0answers
27 views

Divided difference [closed]

$[1, 3, ... 101; x^{52}]$ - divided difference of the function $X^{52}$ on the points 1, 3 ... 101 I reach the point where it equals $S_1^2 - S_2$ where $S_1$ and $S_2$ are Vieta's polynomials. ...
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3answers
68 views

Does a sequence of moments determine the function?

Related questions and answers: Find a smooth function with prescribed moments When do equations represent the same curve? Consider a real valued integrable function $f(x)$ at the interval $a \le x ...
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1answer
286 views

Reconstruct Control points in a Bézier Curve?

I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically ...
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2answers
20 views

How to determine the smallest interpolation degree required?

Given a set of $n$ points $(x_k, y_k)\ (k\in\{1,...,n\})$, of course a polynomial of degree $n$ can fit all points. However, in some cases the coefficient of the higher degrees actually vanish and one ...
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0answers
678 views

How to derive Hermite polynomial from a given data set?

The problem is asking to find a Hermite polynomial to predict the position of the car and its speed when t = 10s. The Hermite polynomial formula is defined as: $$H_{2n+1}(x) = f[z_0] + ...
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103 views
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1answer
32 views

Developing a function of two variables from given data

Cross listed with Mathematica SE: http://mathematica.stackexchange.com/questions/66086/developing-a-function-of-two-variables-from-given-data I have been stuck on the following problem. Consider a ...
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1answer
17 views

Nearest-neighbor interpolation

I read in a book that the nearest-neighbor interpolation results in a function whose derivative is either zero or undefined. Can anyone explain what does it mean when the derivative of a function is ...
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1answer
16 views

Interpolation in quadtrees/octrees

I'm looking for an interpolation algorithm for quadtrees and octrees that is derived from bi(tri)linear or bi(tri)cubic interpolation. I'm mostly interested in the case where: the interpolant is ...
2
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1answer
2k views

Interpolation of a logarithmic function

I have a logarithmic function $$m \ln(x) + b$$ And three points $$(x_0, y_0), (x_1, y_1), (x_2, y_2)$$ The task is to find $m$ and $b$. Do I understand right that the third point is redundant? ...
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0answers
18 views

Application for interpolating periodic B-spline

I need to draw a cubic C^2 continous, closed (periodic boundary conditions) B-spline which should interpolate a set of control points. If possible it would be great if I could specify the knot vector. ...
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2answers
39 views

Second Degree Polynomial Interpolation, error related

We want to create a table of the exponential integral function $$E_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt, x>0$$ over the interval $x \in [1,10]$ with stepsize $h$. How large can $h$ be if a ...
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1answer
22 views

Newton polynomial interpolation degree

8It is asked to find the polynomial of adequated degree to estimate $\sqrt{1.035}$. The following table is given: We know that 1.03 and 1.04 need to be used.Calculation the divided differences ...
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2answers
26 views

Finding an algorithm to mark a lens barrel

I have a zoom lens that only has a handful of focal lengths marked on the zoom ring. I want to make some intermediate marks, but I don't know the math required. I do have the approximate angles of the ...
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1answer
41 views

Finding the value of y using Lagrange Formula

Let $p_2(x)$ be the interpolating polynomial for the data $(0 , 0) , (0.5 , y) , (1,3)$ from Lagrange formula. The coefficient of $x^2$ in $p_2(x)$ is $-2$ , Find the value of $y$ .
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0answers
24 views

Interpolation between two points

I am looking for an interpolation between two points $P$ and $Q$. I need the curve to have derivative (direction) $\vec{v_1}$ at point P and $\vec{v_2}$ at point Q. In addition, there is a maximum ...
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0answers
18 views

Interpolating between many inputs and two outputs

We have a piece of computer software that we need to estimate the minimum requirements for. The requirements will be parametrized by certain usage factors, and expressed in terms of CPU and memory ...
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0answers
17 views

How to interpolate multidimensional functions?

I'm learning about interpolation and I wanted to ask if there's a "good" method to interpolate multidimensional functions (when the dimension can be even a few thousands)? Is there a theoretic limit ...
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1answer
32 views

Transforming $[-3, -2, -1, 0, 1, 2, 3]$ to $[0.125, 0.25, 0.5, 1, 2, 4, 8]$

Given the range of negative/positive numbers $[-3, -2, -1, 0, 1, 2, 3]$, is there a transformation that gives me $[0.125, 0.25, 0.5, 1, 2, 4, 8]$?
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1answer
25 views

Interpolation polynomial types

I was wondering if both the Maclaurin and Taylor series are two types of interpolation polynomials? I was under the impression that they were not because they only go though one point in an interval ...
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2answers
38 views

Cubic spline solving equation

$$S(x)=\begin{cases} x^3 +4x^2 -2x +7 & \text{ if } -1\leq x\leq 0, \\ x^3 - 2x^2 +4x +5& \text{ if } 1\leq x\leq 2, \end{cases}$$ is a cubic spline with knots $\{-1, 0, 1, 2\}$ ...
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3answers
56 views

Why do we choose cubic polynomials when we make a spline?

Good morning, I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work. To me ...
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0answers
30 views

Cubic polynomial interpolation

Let $f(x) = x^2\cdot (x-1)^2 \cdot (x-2)^2 \cdot (x-3)^2$. What is the piecewise cubic Hermite interpolant of $f$ on the grid $x_0 = 0$, $x_1 = 1$, $x_2 = 2$, $x_3 = 3$. Let $g(x) = ax^3 + bx^2 + cx ...
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0answers
15 views

Error of linear Interpolation with intermediate points obtained from an explicit RKM

For the initial value problem $y'(t)=f(t,y(t))$ $(f\in C^\infty(\mathbb{R^2}))$ with $t\in [a,b]$ and $y(a)=y_0$ let $u_k, k=0,...,n$ be the approximation of $y(t_k)$ obtained from an explicit ...
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0answers
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divided diferences

Let $f(x)=\frac 1x$ and prove that $f(x_0,x_1,\ldots,x_n = (-1)^n \prod_{i=0}^{n}{x_i}^{-1}$. i can clearly show that for k=0 $f(x_0) = (-1)^0 \prod_{i=0}^{0}{x_0}^{-1}$. = 1/x0 but how do i show ...
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1answer
60 views

Calculating coefficients of interpolating polynomial using Neville's algorithm

First of all, sorry for my bad math terminology as it's not my native language and I may misuse some terms in English. I've been tasked with writing an application which calculates the general ...
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1answer
278 views

Spline with varing tension, selection of tension factor

I need to perform a special interpolation, using that kind of basis : $$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$ where the $a_i$, $b_i$, $c_i$ and ...
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2answers
27 views

Lagrange interpolation for rational functions

Lagrange interpolation is very useful. I was wondering if there was an equivalent that is not using polynomials but rational functions, one polynomial divided by another. Look at this example: Say I ...
2
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1answer
601 views

Akima spline interpolation

I want to use Akima interpolation on series of points. I have those points in 3D [x, y, z]. But in all resources, I found, there is only f(x) and x (so [x,y]). In Natrual Cubic Spline I am using this ...
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1answer
272 views

Wolfram Mathematica - Newton Backward Interpolation?

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
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1answer
42 views

I am not understanding this step

I am starting the chapter on differential equations and have this example to work through but I do not understand a few things Solve $dy=\frac{dy}{dx}=\frac{2x(y-1)}{x^2+1}$ solution: note that ...
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0answers
15 views

MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
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0answers
29 views

In interpolation, why does my choice of $x_0…x_n$ matter?

This is more of a theoretical question regarding my choice of x's for my interpolation. I'm wondering if someone can explain to me why when I choose different x's for approximating a value at a point, ...
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0answers
15 views

Merging two univariate functions gracefully

Suppose I tell you that $$ f(0) = 0 $$ $$ f'(0) = 0 $$ and $$ f''(0) = a $$ for known $a>0$, whereas for large $x$ $$ f'(x) \approx \cosh^{-1}(x) $$ for $x>2$. Knowing nothing else ...
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0answers
14 views

Calculate (designate) area of ​​the largest area

The values ​​of a function of two variables z = f (x, y), where x, y, z are float. Calculate (designate) area of ​​the largest area of flat O. By 'flat area' O mean a sub area T wherein for each ...
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0answers
13 views

Calculate (designate) area of ​​the largest area

The values ​​of a function of two variables z = f (x, y), where x, y, z are float. Calculate (designate) area of ​​the largest area of flat O. By 'flat area' O mean a sub area T wherein for each pair ...
0
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0answers
30 views

What is it called when we interpolate a point INTO a grid…

Consider a uniform 2D grid, where each $(x,y)$ value on this grid has a corresponding value. So, if I want to find the value, $v$ (unknown) of a point that exists at some arbitrary co-ordinate $(x,y)$ ...
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1answer
526 views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
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1answer
22 views

Existence of function with prescribed values?

Does there exist an infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ equal to $|x|$ when $x \in \mathbb{Z}$?
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17 views

What does the Stein–Weiss Interpolation Theorem say?

I was looking for the statement for the Stein–Weiss Interpolation Theorem, but I cant find it anywhere on internet.
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10 views

Polynomial Interpolation Existence and Uniqueness

The question I am attempting to solve is as follows: Let $f$ be a polynomial of degree $\le n$ and let $p_n$ be a polynomial interpolant to $f$, at the $n+1$ distinct nodes $x_0,x_1,...,x_n$. PROVE ...
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0answers
17 views

Rational function interpolation?

We know that $n+1$ points is enough to completely determine a polynomial of degree $n$. Are there any techniques which says that a certain number of points is enough to completely determine a rational ...
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0answers
30 views

Change of basis from Chebyshev to monomial basis for polynomials

I'm not that familiar with Chebyshev polynomials, so I hope I'm not too far off. Suppose that I have three order pairs $(x_0, f(x_0))$, $(x_1, f(x_1))$, and $(x_2, f(x_2))$ where $f : \mathbb{R} \to ...
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1answer
50 views

Determine error in Neville's Algorithm calculation

I've been mulling over this problem for a while and I don't even know how to start it. The book is hopelessly vague. The problem states Neville's Algorithm is used to approximate $f(0)$ using ...