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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Bound for Lagrange interpolant

I'm having difficulty figuring out how the bound in the below example is determined. Consider the function $f(x)=e^{-x}$. For sample points $\{-1,1\}$, the Lagrange basis interpolants are ...
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28 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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Exponential interpolation using values from exponential distribution [on hold]

I have to approximate an exponential decreasing curve, $ae^{-ax}$ which are the formula to do it using $x$ and $y$ values of the points available?
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11 views

Empirical Mode Decomposition: How to make a suitable spline interpolation when the number of extrama is small?

I am doing an EMD(Empirical Mode Decomposition) project and I am getting a problem with the spline step (find the upper and lower envelopes) because there is always a big deviation in spline ...
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20 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 ...
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2answers
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Interpolation using rate of change

I have a set of data with missing points, which I estimated using spline interpolation. I've now been given the rates of change at each data point. How will this change/improve my current ...
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4answers
129 views

Find a function with certain requirements

I'm trying to find a function $y=f(x)$ that can be described as follows: $f(x) = g(x) + c/(x-x_a)$. With $f(x)$ I want to design a function with the following properties: $f(0) = 0$; $f(x)$ has a ...
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26 views

Help to find the best lower bound function for a given set of data, based in the natural logarithm function

I am trying to find a lower bound function for a set of data I have, and I am struggling with it. In the following graph the blue color is the set of data and the red color is my lower bound function. ...
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27 views

Mathematics and Algorithms for Interpolation

I am doing some programming, where I am interpolating point a to point b, against a timer that is constantly incrementing by ...
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18 views

Combining samples for interpolation

I'm writing a program where values at a position in a 3D field should be estimated, based on a number of existing samples. In this case it is the density of a point cloud at different positions in ...
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18 views

Increase number of points after interpolation

I have a set of points which would give a curved line.I want to increase the number of points on the curve. eg- In the diagram, the original points are shown in highlighted black. If I interpolate ...
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9 views

Calculating 4 components (W,G,R,B) via interpolation

I am trying to find a formula for calculating the 4 components via some type of interpolation. Being very new to the subject, never studying it, I would appreciate if anyone could point me in the ...
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13 views

Healing a curtailed function

Suppose we have a function f : $\mathbb{R} \rightarrow \mathbb{R}$ and a ceiling M. We could define a second function $ g(x) = \left\{ \begin{array}{lr} f(x) & : f(x) < M \\ ...
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creating polynomials using sets of data to represent a correlation [migrated]

I need some guidance in creating a polynomial function that represents sets of data and its correlation, if that makes any sense. I know there Lagrange interpolation, least squares etc. I don't know ...
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1answer
29 views

How can I approximate a function that is not derivable with derivable ones?

Suppose that I have a function whose graph has many angles (i.e. my function is not derivable). How can I approximate this function with derivable ones? Thank you!
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Find $a,b,c$ of polynomial function (Hermite interpolation)

Given abscissae $x_1 < x_2 < \dots < x_N$ and corresponding data values $\{y_i\}_{i=1}^N$ and derivative values $\{y_i'\}_{i=1}^N$, consider the following Hermite interpolation method: For ...
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1answer
30 views

What to do when the chebyshev point is equal to data point in lagrange interpolation?

I am going to use Lagrange interpolation using Chebyshev nodes using the following formula $$\sum_x \prod_{k=0,k\not={j}}^n \frac {x-y_k}{y_j-y_k} f(x) $$ in which $x$ in my data points, $y_k $s ...
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56 views

proving linear interpolation of Level Set

I tried to explain figure below in mathematics form. As you can see I have got triangle (v1, v2, v3). The signed shortest distance form red interface (level set value) is calculated for each vertex ...
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39 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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1answer
20 views

Upper bound for the error magnitude

for the function $f(x) = e^x$ on the interval [0,1] by using polynomial interpolation with $x_0 = 0, x_1 = 1/2, x_2 = 1$ find the upper bound for the magnitude $\max_{0 \leq x \leq 1} |e^x ...
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1answer
24 views

Find $f(x)$ given $f(0), f(1)$ and $f[x1,x2,x3]$

I need to find f(x) given $f(0) = 0$, $f(1) = 2$, and the divided difference $f[x_1,x_2,x_3] = 1$ for any three points $x_1, x_2, x_3$ How do I go about solving this?
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9 views

Using linear interpolation between two points to find the three remaining points

I am taking a graphics programming course, and I am looking at how Linear interpolation can be used to move points from one location to another location within a certain time. My mathematics ...
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74 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 ...
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75 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 ...
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19 views

Partition of function into pieces for interpolation needs

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I ...
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using approximation to find interpolantion function?

my question is can we actually find the interpolating polynomial if we solve the approximation problem for degree m = n-1 ( where we have n data ).(i know we usually solve the approximation problem ...
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11 views

Recovering a continuous density function from its discretized version.

The probability density function is defined on [0,10], and it's discretized by taking integration over short intervals [0,0.01],[0.01,0.02], etc. Is there any kind of interpolation method ( I'm not ...
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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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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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1answer
28 views

deriving an integral quadrature rule on a triangle

I'm trying to look for references on this but I've not found any. I'm probably using the wrong keywords ... Let's suppose that our domain of integration $\Omega$ is the triangle in $R^2$ with a ...
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1answer
48 views

Lagrange Interpolating Polynomials - Error Bound

Let $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the ...
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23 views

Best fit in logarithmic chart

I have several variances ($\sigma^2$) which value depends on the velocity ($v$). As you can see in the graph, if increase the velocity, the variance does the same. I am studying this dependency, but ...
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35 views

Newton form of the interpolation polynomial?

Let $ f(x)=\sin(x) $ and let $ p_1 $ be the first degree polynomial that interpolates $ f $ at $0$ and $\pi/2$. Then $ p_1(x)=(2/pi)x. $ How did they get this result; how is $ p_1(x) $ found?
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Prove that the difference between a continuous function and its interpolation polynomial..?

Let $ g \in C^2 [x,y] $ and $ P $ be its interpolation linear polynomial at $ a_0 $ and $ a_1 $ in $ [x,y] $. Prove that $ \lVert g-P\rVert _\infty < 1/8(a_1-a_0)^2B $ where $ \lvert ...
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38 views

Natural Cubic Spline Unequal Spacing

I'm currently trying to create a natural cubic spline by setting up linear equations to solve for the coefficients of the spline. Where the spline follows as : ...
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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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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 ...
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What equation would be used to be draw a curve like this?

I have created a little visual based programming system and am not working on the visuals, if two nodes are connected and B.x<A.x then I want a curve to be drew in the fashion above.
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Convergence of quadrature formulas and interpolating polynomials

There is a theorem of Polya (1933), which says: 1) If a interpolatory quadrature formula converges for all continuous functions on [a, b] and quadrature weights are all positive, then the formula ...
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33 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 ...
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Find nonlinear growth formula out of a set of values

I got two sets of values: $(x=240,y=20)$ and $(x=960,y=480)$. How can I find a formula to get any values in between? Anyone knows? Also if I distribute the value of $x$ among subvalues (imagine $x$ ...
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68 views

Newton's forward-difference formula question?

Use Newton's forward-difference formula to construct interpolating polynomials of degree two, and three for the following data. Approximate the specified value using each of the polynomials. I ...
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21 views

Interpolation with a constrained range between given control points

I am trying to create an algorithm that creates smooth color gradient functions, given control points in the red, green, and blue components. Mathematically, each curve would have a domain [0, 1] ...
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Lagrange polynomials question

Construct the Lagrange interpolating polynomials for the following functions, and find a bound for the absolute error on the interval $[x_0, x_n]$. $f(x) = e^{2x}\cos3x, ~~~~~~~x_0=0, ~~~~~~~x_1=0.3, ...
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82 views

Possible generalizations of Hadamard's three line lemma

Let $f$ be an analytic function on a sector $$ S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\} $$ with opening angle $\gamma$ at the origin. Suppose $f$ is ...
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Linear interpolation vs polynomial interpolation

Why linear interpolation is better than polynomial interpolation when we want to approximate $f(0.25)=e^{0.25}$? I can't formulate a concrete explanation. I thought that maybe it has a link with the ...
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Estimate the error of the Lagrange Interpolating polynomial

consider the function, f(x) = 1/(1+x^2) Estimate the error of the Lagrange Interpolating polynomial of 5 equally spaced points in the interval [-5,5]. Is there a proper way to find this logically ? ...
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86 views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
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32 views

Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...
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Finding the period of the function

My question is as follows:- Find the trigonometric interpolant $ \bar{f}(x)$ for $f(x)= \frac{\pi}{x+3\pi}$ and $n=1$. Thant is, find coefficients $c_{-1}, c_0 ,c_1$ such that ...