# Tag Info

### What is the preferred and efficient approach for interpolating multidimensional data?

For the first part of my question, I found this very useful comparison for performance of different linear interpolation methods using python libraries: http://nbviewer.ipython.org/github/pierre-...
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### Example of a continuous function that is difficult to approximate with polynomials

Why not simply show the absolute value function? Approximation with e.g. Legendre-polynomial expansion works, but pretty badly: Taylor expansion is of course completely useless here, always giving ...
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### Interpolation by Solving a Minimization Problem (Optimization)

This is the classic colorization using optimization problem. Optimization and Linear Algebra To see how this can be expressed as a linear system, it's helpful to use a slightly different notation (...
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### Example of a continuous function that is difficult to approximate with polynomials

It's a pathological case, but you can always resort to the Weierstrass monster function. It illustrates a broader point, namely that functions that are not smooth -- e.g., that have a kink -- are ...
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### Linear interpolation in Fortran

There is no built-in Fortran functionality to do linear interpolation. You could either use a library or write your own routine. I haven't tried compiling or testing and my fortran may be a bit rusty, ...
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### Interpolating a mathematical function using a Hermite Cubic Finite Element Space

Short answer You are missing the Jacobian of the transformation for the derivatives. Long answer The conditions that you propose for your interpolator translate into the following system of ...
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### Approximating a step function with polynomials

This problem is generally called the Minimax Problem. Unfortunately the step function is not continuous and therefore the Weierstrass approximation theorem does not apply. Any continuous approximation ...
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### Interpolate the orbital coordinates of an object using coordinates and velocities vector

That depends on how well you know the coordinates and velocities. If you have exact values, you can get a reasonable answer using Hermite interpolation. This will give you a degree-3 polynomial in ...
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### Error of interpolating polynomial

For a polynomial interpolant we have the formula for the pointwise error $$E_n(x) = f(x) - p_n(x) = \frac{f^{(n+1)}}{(n+1)!}\prod_{i=0}^n(x-x_i),$$ where the $x_i$ are the interpolation knots. In ...

### Fourier transform by FFT : by using cubic splines to interpolate between data points, do we change the frequency content of the Fourier transform?

The frequency content of the interpolated signal is significantly influenced by the interpolation basis. If you have a band-limited function that you have adequately sampled (i.e. satisfying Nyquist ...
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### Parametrized spline - oscilating second derivative

Your choice of parameterization is creating problems. Instead of spanning one in $t$ between points, span an amount proportional to the line segment between the two points in $(x,y)$ space. I've ...
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### What are the benefits of using machine learning for interpolation over traditional interpolation methods?

First of all, interpolation and approximation are slightly different from each other. Given a sufficiently smooth function $f$ (sufficiently smooth just means that I am covering my bases, there are ...
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### What is the difference between $u_h$ and $I_h(u)$ in finite element literature?

The other answer has everything you already need, but it's also worth pointing out that $u_h$ is computable whereas $I_hu$ is not: The latter requires you to know the exact solution, which in general ...
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### Interpolation with the roots of orthogonal polynomials & Spectral expansion

I hope I understood the question correctly. They try to compute exactly the same thing, so they really are equivalent. I'll use Chebyshev polynomials because they are easy to analyze. Given a ...
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