A Multiresolution Analysis (MRA) of L^{2}(**R**) is a nested family of subspaces,

... ⊆ V_{-1} ⊆ V_{0} ⊆ V_{1} ⊆ V_{2} ⊆ ...

such that

i) If f(x) is in V_{n} then f(2x) is in V_{n+1},

ii) V_{0} contains integer translations of its elements, (i.e. if f(x) is in V_{0}, then so are f(x-k) for any integer k.)

iii) There is a function φ(x) in V_{0}, called the scaling function, which "generates" the space V_{0} - in the sense that, any function in V_{0} can be expressed as a linear combination of the functions {φ(2x-k)} (for integer k).

iv) The intersection of the spaces {V_{i}} is equal to {0}.

v) The union of the spaces {V_{i}} is dense in L^{2}{**R**).

Because of (i) and (ii), as you go higher up in the spaces V_{n}, you will capture "finer detail".

As a consequence of (iii), φ itself can be expressed as a linear combination of compressed translations. i.e. φ(x) = Σ_{k} s_{k} φ(2x-k) for some real (or complex) coefficients {s_{k}}. These are called the scaling coefficients. A function satisfying such an identity is often called self similar. The equation is similar to, but not quite, a finite difference equation.

If the scaling function φ is orthogonal, meaning that the L^{2 }inner product <φ(x-k), φ(x-j)> = ∫ φ(x-k) φ(x-j) dx = δ_{jk }, this is sometimes called an orthogonal MRA. If the scaling function is compactly supported, there will be finitely many non-zero scaling coefficients, and vice versa. Often times the scaling function will be continuous, or of finite support, or twice differentiable, in order to grant the MRA nice features.

In this orthogonal case, one could define the wavelet space W_{n} as the orthogonal complement of V_{n} within V_{n+1}. A wavelet is formed (as a linear combination of the {φ(2x-k)}) by using the wavelet coefficients, and these are defined by reversing the order of the scaling coefficients and alternating the sign on each coefficient. (w_{k} = (-1)^{k} s_{1-k}). This wavelet, being an element of W_{0}, will be orthogonal (in the L^{2} sense) to all of V_{0}, as it is orthogonal to the scaling function φ(x) which generates V_{0}.

A good first example of an orthogonal scaling function, despite not being continuous, is the Haar function. This function is the identity (equal to 1) over the interval [0, 1], and it is equal to 0 elsewhere. The scaling coefficients would be {1, 1}, because given the Haar function as p(x), we would have p(x) = p(2x) + p(2x-1).

The Haar wavelet would use the wavelet coefficients {1, -1}, it would be defined as ψ(x) = p(2x) - p(2x-1), and it would be equal to one over [0, 1/2], and equal to negative one over [1/2, 1]. (You can see how integrating this times the Haar function would give you zero - they are orthogonal.)

A more interesting example, though not orthogonal, is the "tent" function T(x), the piecewise linear function increasing from 0 to 1 over the interval [-1, 0], then decreasing from 1 to 0 over the interval [0, 1]. The scaling coefficients in this case are {1/2, 1, 1/2}, as the function T(x) = 1/2 T(2x+1) + T(2x) + 1/2 T(2x-1).