The transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet

Wavelet

A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...

 theory and finite element theory.

For the mask , which is a vector with component indexes from to ,
the transfer matrix of , we call it here, is defined as
More verbosely
The effect of can be expressed in terms of the downsampling

Downsampling

In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....

 operator "":

Properties

• .
• If you drop the first and the last column and move the odd indexed columns to the left and the even indexed columns to the right, then you obtain a transposed Sylvester matrix
  Sylvester matrix
  In mathematics, a Sylvester matrix is a matrix associated to two polynomials that provides information about those polynomials. It is named for James Joseph Sylvester.-Definition:...
  
  .
• The determinant of a transfer matrix is essentially a resultant.

More precisely:

Let be the even indexed coefficients of () and let be the odd indexed coefficients of ().

Then , where is the resultant.

This connection allows for fast computation using the Euclidean algorithm

Euclidean algorithm

In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...

.

• For the trace
  Trace (linear algebra)
  In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
  
   of the transfer matrix of convolved
  Convolution
  In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
  
   masks holds
• For the determinant
  Determinant
  In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
  
   of the transfer matrix of convolved mask holds

where denotes the mask with alternating signs, i.e. .

• If , then .

This is a concretion of the determinant property above. From the determinant property one knows that is singular whenever is singular. This property also tells, how vectors from the null space

Null space

In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space...

 of can be converted to null space vectors of .

• If is an eigenvector of with respect to the eigenvalue , i.e.

,

then is an eigenvector of with respect to the same eigenvalue, i.e.

.

• Let be the eigenvalues of , which implies and more generally . This sum is useful for estimating the spectral radius
  Spectral radius
  In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ.-Matrices:...
  
   of . There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small .

Let be the periodization of with respect to period . That is is a circular filter, which means that the component indexes are residue classes with respect to the modulus . Then with the upsampling

Upsampling

Upsampling is the process of increasing the sampling rate of a signal. For instance, upsampling raster images such as photographs means increasing the resolution of the image....

 operator it holds

Actually not convolutions are necessary, but only ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the Fast Fourier transform

Fast Fourier transform

A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

.

• From the previous statement we can derive an estimate of the spectral radius
  Spectral radius
  In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ.-Matrices:...
  
  of . It holds

where is the size of the filter and if all eigenvalues are real, it is also true that,

where .