kaizen.moto said:How about f(x) = (1 - x/a), what would be the solution after Fourier expansion?
A Fourier expansion, also known as a Fourier series, is a way to represent a periodic function as a sum of sine and cosine waves. It is named after French mathematician Jean-Baptiste Joseph Fourier, who first introduced the concept in the early 19th century.
A Fourier expansion is useful in mathematics because it allows us to approximate any periodic function with a finite number of terms. This makes it easier to analyze and manipulate complex functions, as well as solve differential equations and other problems.
A Fourier expansion consists of a series of coefficients, known as Fourier coefficients, and a set of basis functions, typically sine and cosine waves. The coefficients determine the amplitude and phase of each basis function, which together form the periodic function being represented.
No, a Fourier expansion can only be used for periodic functions. However, there are other methods, such as the Fourier transform, that can be used to represent non-periodic functions as a sum of sinusoidal functions.
While Fourier expansions are a powerful tool in mathematics, they do have some limitations. For example, they may not converge for functions that are discontinuous or have sharp corners. In addition, they may not provide an accurate representation for functions with rapidly changing behavior.