# Chebyshev Spectral Decomposition

You won't believe how powerful it isChebyshev Spectral Decomposition and Chebyshev Interpolants are based on a key theorem by Bernstein which states that, for **analytical functions**, Chebyshev Spectral projections and interpolants **converge exponentially** on the number of points where we know the value of the function. This theorem is outstanding; typically used approximation frameworks tend to converge linearly.

This means that we can approximate any analytical function, using Chebyshev, with **extreme accuracy** (e.g. 10E-5) with knowing the value of the function in very few points (e.g. 10 points). Once the approximating objects are created, evaluating them is very efficient. Typically, **one million pricings can be obtained in less than a second** on a standard PC.

Importantly, these results are **mathematically proven**, so they pass all model validation and regulatory constraints.

This theory can be extended to multidimensional functions, as well as to the derivatives of the functions (i.e. its sensitivities).

As a result, it is an ideal mathematical framework to build methods that are very fast-to-compute versions of any analytical function.

Often, the computational bottleneck of risk calculations is calling the same portfolio pricing functions lots of times; **pricing functions are always analytical** (or piece-wise analytical) functions… **and this is where the MoCaX story commences**.

# Resources

If you want to research this fascinating field by yourself, we can recommend

- Our research papers in this field, with applications
- A very good monograph on this topic, “Approximation Theory and Approximation Practice”, Siam, by Prof. L.N. Trefethen (Amazon link)
- chebfun.org, an open source project in Matlab
- Kathrin Glau research

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