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We revisit the construction of wavelets on the interval with various degrees of polynomial exactness, and explain how existing schemes for orthogonal- and Spline wavelets can be extended to compactly supported delay-normalized wavelets. The contribution differs substantially from previous ones in how results are stated and deduced: linear algebra notation is exploited more heavily, and the use of sums and complicated index notation is reduced. This extended use of linear algebra eases translation to software, and a general open source implementation, which uses the deductions in this paper as a reference, has been developed. Key features of this implementation is its flexibility w.r.t. the length of the input, as well as its generality regarding the wavelet transform.

In this chapter we consider LU factorizations of Hermitian and positive definite matrices. Recall that a matrix \({\boldsymbol {A}}\in {\mathbb {C}}^{n\times n}\) is Hermitian if A∗ = A, i.e., \(a_{ji}=\overline {a}_{ij}\) for all i, j. A real Hermitian matrix is symmetric. Since \(a_{ii}=\overline {a}_{ii}\) the diagonal elements of a Hermitian matrix must be real.

In this chapter we first review Gaussian elimination. Gaussian elimination leads to an LU factorization of the coefficient matrix or more generally to a PLU factorization, if row interchanges are introduced. Here P is a permutation matrix, L is lower triangular and U is upper triangular.

The factorization is easily verified. Here \( \mathit\mathbf{U}=\mathit\mathbf{V}=\frac{1}{\sqrt{2}} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right], \)which is clearly unitary. Since in addition the matrix in the middle is diagonal, it follows that this is both a spectral and a singular value decomposition.

Matrices arising from 2D and 3D problems sometimes have a Kronecker product structure. Identifying a Kronecker structure can be very rewarding since it simplifies the derivation of properties of such matrices.

Given a symmetric positive definite matrix \( \mathit\mathbf{A}\epsilon\mathbb{R}^{n \times n} \) with eigenvalues \( 0 < \lambda_{\mathit n} \leq \cdots \leq \lambda_{1} \).

In this chapter we present a fast method for solving Ax = b, where A is the Poisson matrix ( 10.8).

The QR algorithm is a method to find all eigenvalues and eigenvectors of a matrix. In this chapter we give a brief informal introduction to this important algorithm.

In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a QR decomposition and a reduced form which we refer to as a QR factorization. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.

No, we have seen that 4n3/3 operations are required in order to bring a matrix to upper triangular form using Householder transformations.

The conjugate gradient method was published by Hestenes and Stiefel in 1952, as a direct method for solving linear systems. Today its main use is as an iterative method for solving large sparse linear systems. On a test problem we show that it performs as well as the SOR method with optimal acceleration parameter, and we do not have to estimate any such parameter. However the conjugate gradient method is restricted to positive definite systems. We also consider a mathematical formulation of the preconditioned conjugate gradient method. It is used to speed up convergence of the conjugate gradient method. We only give one example of a possible preconditioner.

In this introductory chapter we give a compact introduction to linear algebra with emphasis on \({\mathbb {R}}^n\) and \({\mathbb {C}}^n\).

In this chapter we consider three problems originating from: cubic spline interpolation, a two point boundary value problem, an eigenvalue problem for a two point boundary value problem.

In Chap. 1 we saw how a periodic function can be decomposed into a linear combination of sines and cosines, or equivalently, a linear combination of complex exponential functions. This kind of decomposition is, however, not very convenient from a computational point of view. The coefficients are given by integrals that in most cases cannot be evaluated exactly, so some kind of numerical integration technique needs to be applied. Transformation to the frequency domain, where meaningful operations on sound easily can be constructed, amounts to a linear transformation called the Discrete Fourier transform. We will start by defining this, and see how it can be implemented efficiently.

In Chap. 5 we expressed wavelet transforms, and more generally filter bank transforms, in terms of filters. Through this one obtains intuition on a how a wavelet transform splits the input using low-pass and high-pass filters, and how the filters in the MP3 standard split the input into frequency bands.

Previously we have associated several MRA’s with filter bank transforms. Since filter bank transforms also can be constructed outside the setting of wavelets, it would be interesting to see when we can make the association the opposite way: Which filter bank transforms appear in the context of a multiresolution analysis? An answer to this question certainly could transfer much theory between wavelets and filters. Also, it may be easier to construct good filter bank transforms than good wavelet bases.

We saw in the previous chapter that the wavelet kernels G and H had some repeating structure in the rows and columns, similar to the circular Toeplitz structure in filters. Clearly the matrices are not filters, however. Nevertheless, we will now prove that wavelet kernels can be implemented easily in terms of filters. but that several filters are needed in the computation. Each of these filters will have an interpretation in terms of how the wavelet transform treats different frequencies. Much has been done in establishing efficient implementations of filters, and by expressing a wavelet transform in terms of filters we can take advantage of this.

A major part of the information we receive and perceive every day is in the form of audio. Most sounds are transferred directly from the source to our ears, like when we have a face to face conversation with someone or listen to the sounds in a forest or a street. However, a considerable part of the sounds are generated by loudspeakers in various kinds of audio machines like cell phones, digital audio players, home cinemas, radios, television sets and so on. The sounds produced by these machines are either generated from information stored inside, or electromagnetic waves are picked up by an antenna, processed, and then converted to sound.

In the first part of the book the focus was on approximating functions or vectors with trigonometric counterparts. We saw that Fourier series and the Discrete Fourier transform could be used to obtain such approximations, and that the FFT provided an efficient algorithm. This approach was useful for analyzing and filtering data, but had some limitations. Firstly, the frequency content is fixed over time in a trigonometric representation. This is in contrast to most sound, where the characteristics change over time. Secondly, we have seen that even if a sound has a simple trigonometric representation on two different time intervals, the representation as a whole may not be simple. In particular this is the case if the function is nonzero only on a very small time interval.

We state limit distribution results for random matrices with independent or free entries, also addressing when we get freeness in the limit and semicircular and circular limits. The results generalize some already known results about asymptotic freeness of large random matrices, but our goal is to get a more optimal flavour on these results. When having matrices with identically distributed entries, we show that freeness in the limit is natural when we have free entries, but unnatural when the entries are independent, and restricted to the case of circular limits.

We investigate some ways to obtain free families of random variables from an initial free family in a tracial noncommutative probability space. The random variables in this initial free family are multiplied on the left and the right by certain operators from an algebra free from the entire family. Aspects of freeness related to this algebra are also considered.

In this paper, we derive the explicit series expansion of the eigenvalue distribution of various models, namely the case of non-central Wishart distributions, as well as correlated zero mean Wishart distributions. The tools used extend those of the free probability framework, which have been quite successful for high dimensional statistical inference (when the size of the matrices tends to infinity), also known as free deconvolution. This contribution focuses on the finite Gaussian case and proposes algorithmic methods to compute the moments. Cases where asymptotic results fail to apply are also discussed., 14 pages, 13 figures. Submitted to IEEE Transactions on Information Theory

The distribution of randomly deployed wireless sensors plays an important role in the quality of the methods used for data acquisition and signal reconstruction. Mathematically speaking, the estimation of the distribution of randomly deployed sensors can be related to computing the spectrum of Vandermonde matrices with non-uniform entries. In this paper, we use the recent free deconvolution framework to recover, in noisy environments, the asymptotic moments of the structured random Vandermonde matrices and relate these moments to the distribution of the randomly deployed sensors. Remarkably, the results are valid in the finite case using only a limited number of sensors and samples.