Frequency representations
Wavelet bases

First of all, we need to clarify the meaning of the word "tool". Today's mathematicians have a multitude of techniques for analyzing, synthesizing and representing any number of functions using elementary "building blocks". These techniques for harmonic analysis in the broadest sense are sometimes combined with high-performance algorithms, making them even more useful for numerical applications.

The most fundamental example is certainly the Fourier transform, known since the 19th century: this consists first of all in performing a frequency analysis of a function f (t ), $t\in ℝ$ by the formula :