Basis function

Basis function is a term used in various areas of mathematics, including linear algebra, calculus, and functional analysis. It refers to a set of functions that are linearly independent and can be used to represent any function in a particular function space.

Definition

In the context of linear algebra, a basis function is a function that belongs to a set of functions that are linearly independent and span a function space. This means that any function within that space can be represented as a linear combination of the basis functions.

In functional analysis, a basis function is often used to construct a series that converges to a given function. The most common examples of this are the Fourier series and the wavelet series, where the basis functions are sinusoids and wavelets, respectively.

Properties

Basis functions have several important properties. They are:

• Linearity: The linear combination of basis functions is also a basis function.
• Independence: No basis function can be represented as a linear combination of the other basis functions.
• Completeness: Any function in the function space can be represented as a linear combination of the basis functions.

Applications

Basis functions are used in a wide range of applications, including:

• Signal processing: In signal processing, basis functions are used to decompose signals into simpler components. This is often done using the Fourier transform or the wavelet transform.
• Machine learning: In machine learning, basis functions are used to transform input data into a higher-dimensional space, making it easier to learn complex patterns. This is often done using kernel methods.
• Quantum mechanics: In quantum mechanics, basis functions are used to represent the state of a quantum system. This is often done using wave functions.

See also

• Linear combination
• Function space
• Fourier series
• Wavelet series