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applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes
refinable function In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function \varphi is called refinable with respect to the mask h if :\varphi(x)=2\cdot\sum_^ h_k\cdot\varphi(2\cdot x-k ...
s. Refinable functions play an important role in
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
theory and
finite element The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...
theory. For the mask h, which is a vector with component indexes from a to b, the transfer matrix of h, we call it T_h here, is defined as : (T_h)_ = h_. More verbosely : T_h = \begin h_ & & & & & \\ h_ & h_ & h_ & & & \\ h_ & h_ & h_ & h_ & h_ & \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ & h_ & h_ & h_ & h_ & h_ \\ & & & h_ & h_ & h_ \\ & & & & & h_ \end. The effect of T_h can be expressed in terms of the
downsampling In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''com ...
operator "\downarrow": :T_h\cdot x = (h*x)\downarrow 2.


Properties

* T_h\cdot x = T_x\cdot h. * 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 In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant ...
. * The determinant of a transfer matrix is essentially a resultant. :More precisely: :Let h_ be the even-indexed coefficients of h ((h_)_k = h_) and let h_ be the odd-indexed coefficients of h ((h_)_k = h_). :Then \det T_h = (-1)^\cdot h_a\cdot h_b\cdot\mathrm(h_,h_), where \mathrm is the
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...
. :This connection allows for fast computation using the Euclidean algorithm. * For the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
of the transfer matrix of
convolved In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
masks holds :\mathrm~T_ = \mathrm~T_g \cdot \mathrm~T_h * For the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the transfer matrix of convolved mask holds :\det T_ = \det T_g \cdot \det T_h \cdot \mathrm(g_-,h) :where g_- denotes the mask with alternating signs, i.e. (g_-)_k = (-1)^k \cdot g_k. * If T_\cdot x = 0, then T_\cdot (g_-*x) = 0. : This is a concretion of the determinant property above. From the determinant property one knows that T_ is
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
whenever T_ is singular. This property also tells, how vectors from the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...
of T_ can be converted to null space vectors of T_. * If x is an eigenvector of T_ with respect to the eigenvalue \lambda, i.e. : T_\cdot x = \lambda\cdot x, :then x*(1,-1) is an eigenvector of T_ with respect to the same eigenvalue, i.e. : T_\cdot (x*(1,-1)) = \lambda\cdot (x*(1,-1)). * Let \lambda_a,\dots,\lambda_b be the eigenvalues of T_h, which implies \lambda_a+\dots+\lambda_b = \mathrm~T_h and more generally \lambda_a^n+\dots+\lambda_b^n = \mathrm(T_h^n). This sum is useful for estimating the
spectral radius In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
of T_h. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small n. :Let C_k h be the periodization of h with respect to period 2^k-1. That is C_k h is a circular filter, which means that the component indexes are
residue class In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
es with respect to the modulus 2^k-1. Then with the
upsampling In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe a ...
operator \uparrow it holds :\mathrm(T_h^n) = \left(C_k h * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (C_k h\uparrow 2^)\right)_ :Actually not n-2 convolutions are necessary, but only 2\cdot \log_2 n ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the Fast Fourier transform. * From the previous statement we can derive an estimate of the
spectral radius In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
of \varrho(T_h). It holds :\varrho(T_h) \ge \frac \ge \frac :where \# h is the size of the filter and if all eigenvalues are real, it is also true that :\varrho(T_h) \le a, :where a = \Vert C_2 h \Vert_2.


See also

* Hurwitz determinant


References

* * {{cite thesis , first=Henning , last=Thielemann , url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131 , title=Optimally matched wavelets , type=PhD thesis , year=2006 (contains proofs of the above properties) Wavelets Numerical analysis