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Questions tagged [interpolation]

Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points. It is necessary because in science and engineering we often need to deal with discrete experimental data.

Interpolation is a useful mathematical and statistical tool used to estimate values between two points on a line or curve.

What Is Interpolation?

Interpolation is the process of deriving a simple function from a set of discrete data points so that the function passes through all the given data points (i.e. reproduces the data points exactly) and can be used to estimate data points in-between the given ones.

To help us remember what it means, we should think of the first part of the word, 'inter,' as meaning 'enter,' which reminds us to look 'inside' the data we originally had.

Applications: This tool, interpolation, is not only useful in statistics, but is also useful in science, business or any time there is a need to predict values that fall within two existing data points. It is also used to simplify complicated functions by sampling data points and interpolating them using a simpler function. In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate, i.e., estimate the value of that function for an intermediate value of the independent variable.

The details, techniques, and precise meaning of interpolation depend heavily on the sub-discipline of mathematics, therefore you are encouraged to use additional subject tags such as , , , or when appropriate.

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Lagrange's basis function and Interpolation

Let $x_0,...,x_n$ be distinct real numbers and $l_k(x)$ be the Lagrange's basis function. $δ_n = ∏^n _{k=0}(x-x_k)$. Prove that a - $\sum^n_{k=0}x^j_kl_k(x) ≡ x^j$. for $j = 0,1,...,n$ b - Let $P_n(x)$ interpolate $f(x)$ at $x_0,...,x_n$. Then…
user60514
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monotone interpolation given two points and their derivatives

Is there an algorithm or implementation? I have searched the monotone cubic interpolation or monotone piecewise cubic interpolation. It seems that both the two methods cannot preserve the derivatives at the two given points. I am not really sure…
user1446072
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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 $u$.
Vineet Maheshwari
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Lagrange interpolant degree 2

Consider the function $$f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$$ in the interval $I = [-1,1]$. Set $\beta = 0$. The Lagrange interpolating polynomial of $f(x)$ with degree $n=2$ for equally spaced nodes in $I$. I tried several times and I want to…
BRabbit27
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Lanczos Splines

to start I would like to say I do not have much knowledge about this topic. What I would like to know is if it is possible to calculate the blending functions of lanczos resample filter. In: Catmull-Rom splines As it is demonstrated in the bottom of…
Pedro Pereira
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Interpolation with two conditions

I have two towns. For the town1 I have hourly temperatures throughout the whole day for the whole year. For the town2 I have temperatures at 07:00, 14:00, 22:00. That looks something like this: Time Town1 Town2 01.01.2017. 00:00 …
Creakush Coliko
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Stuck at a step in proving $\Delta + \nabla =\dfrac{\Delta}{\nabla} - \dfrac{\nabla}{\Delta}$

$\Delta + \nabla =\dfrac{\Delta}{\nabla} - \dfrac{\nabla}{\Delta}$ L.H.S $= (E-1)+(1-E^{-1}) = E-E^{-1}$ R.H.S $=…
Daman
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Smooth interpolation with discontinuities, irregular grid and different densities.

What would be an appropriate interpolation method that does some smoothing (similar to splines), allows for unknown (relatively big) discontinuities, can be run on a dataset in an irregular grid and with an equally irregular density of points (ie:…
Fidel
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Solving Quadratic Newton polynomial not yielding same solution as divided differences

So a Newton polynomial is defined by $$N(x) = a + b(x-x_0) + c(x-x_0)(x-x_1)$$ Solving this for $x \in \{x_0, x_1, x_2\}$ yields three equations, giving expressions for $a, b, c$: 1: \begin{align*} N(x_0) &= a + b(x_0-x_0) +…
Frank Vel
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What is $\prod_{n+1}(x)$ in the inequality $|R_n(x)|\le \frac{M_{n+1}}{(n+1)!}|\prod_{n+1}(x)|?$

What is $\prod_{n+1}(x)$ in the inequality $$|R_n(x)|\le \frac{M_{n+1}}{(n+1)!}|\prod_{n+1}(x)|?$$ $M_{n+1}$ is defined as $M_{n+1}=\max_{t\in[a,b]}|f^{n+1}(t)|$ where $a=\min\{x,x_0\},b=\max\{x,x_n\}$. What is the definition of $\prod_{n+1}(x)$?
user156262
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Interpolation for Recreating Probable Values Lost During Averaging

I am not a mathematician so my question might appear lame to you all; anyways, here I go. I work with images and here in this problem, I would like to blow up the resolution of the image. Let's say we got a $5\times 5$ pixel matrix with some values…
Arnab Muhuri
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2D interpolation of array of values

I'm trying to interpolate between 2 arrays and I don't know how to this. I have 3 arrays of values: two values array: $[a_1,a_2,a_3, \dots,a_n]$,$[b_1,b_2,b_3,\dots,b_n]$ and each of the arrays has a value $z_1$, $z_2$ respectively. I also have a…
edenbd
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Linear Interpolation?

Just want to ask if this scenario is still possible in linear interpolation. If not, what interpolation would I need to get the missing value: X1 = 50 Y1 = 35 Z1 = 60 X2 = 70 Y2 = 40 Z3 = 70 X3 = 55 Y3 = 30 Z3 = ? Thanks
arcbound08
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Interpolating between two-dimensional splines

My problem can best be described by first looking at my plot: It is very simple I would like to interpolate between the 2D splines that sit atop the coloured faces (they are only there for illustration). Each spline represents time-series…
Astrid
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Bicubic spline interpolation solver

I created a small delphi application to solver bicubic spline interpolation problem. I have a grid of 4x4 functions value and I want use bicubic spline interpolation to get value for x, y point. https://en.wikipedia.org/wiki/Bicubic_interpolation I…
Martin
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