Fourier series are used to solve boundary value problems in partial differential equations. In 1822, Fourier first used this technique to solve the heat equation. A discrete version of the Fourier series can be used in sampling applications where the function value is known only at a finite number of equally spaced points. In this case the Fourier series is finite and its value is equal to the sampled values at all points. The set of coefficients is known as the discrete Fourier transform (DFT) of the given sample sequence. The DFT is one of the key tools of digital signal processing, a field whose applications include radar, speech encoding, image compression. The JPEG image format is an application of the closely related discrete cosine transform.