# Tag Info

### 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 ...
Accepted

### 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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### 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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### 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 ...
• 10.4k

### 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 ...
• 1,031
Accepted

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

The interpolation indeed affects the Fourier transform. @Steve already gives the correct answer in general, but I want to give you an example that helps the intuition more. Think for example that you ...
• 56.2k
Accepted

### 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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