John Schmeelk

Wavelet transforms on two-dimensional images

This paper studies the two-dimensional wavelet transform applied to two-dimensional images. The classical technique oftentimes implements the Fourier transform. This paper offers a brief discussion regarding the comparison of the two transforms on a single alphabet. It provides a comparison of the global properties present in the Fourier transform technique versus a more localized analysis when the wavelet transform is applied to the same image. The wavelet selected in this study is the derivative of the Gaussian since in some sense offers a nice comparison to the Fourier method.

Wavelet transforms and edge detectors on digital images

The paper investigates the problem of locating edges on digital images. The process is completed through various mathematical transforms. The edges are found by implementing mathematical algorithms for convolutions that are adapted to locate the edges and highlights the technique on using wavelet transforms. We are especially addressing the problem of edge detection, because it has far reaching applications to all fields and especially within medicine. A patient can be diagnosed as having an aneurysm by studying an angiogram. An angiogram is the visual view of the blood vessels whereby the edges are highlighted through the implementation of edge detectors. This process can be completed through wavelet transforms and other convolution techniques.

Infinite dimensional parametric distributions

Appropriate definitions and properties are introduced for the scale of Frechet spaces, TPB, and its corresponding dual space defined to be the class of infinite dimensional tempered distributions. A generalization of differentiation is introduced on the class of infinite dimensional tempered distributions. When the real time parameter is introduced into the setting, a new class of infinite parametric distributions are studied as solutions to generalized Schrodinger equations. The class of infinite dimensional exmpered distributions has different representations. It is within the context of their kernel representation that standard creation and annihilation operators are studied. All of these spaces are considered as generalizations of Fock spaces.

An infinite-dimensional Laplacian operator

then F is called the Frtchet derivative. In finite-dimensional vector spaces it is called the directional derivative. The idea of the FrCchet derivative can be extended to both topological and nonnormed pseudotopological linear vector spaces, where it is called a functional derivative. The details of this extension, which is based on the concept of a remainder, can be found in [3]. The classical idea of the gradient can bc defined as follows. Let the primary topological space be a Hilbert space and let g, be a countable complete orthonormal set of vectors. Then the function,

Application of test surfunctions

Appropriate definitions and properties are introduced for the scale of Frechet spaces, . Differentiation of every order is then given for functionals having the space as their domain. These derivatives are similar to Frechetderivatives. For computational considerations thev are evaluated asf We then implement some techniques in our setting to solve abroad class of abstract Cauchy problems. The techniques aresimilar to operator techniques employed to classical problems. The order of the operator is introduced as where λ relates to the usual inverse operator to time differentiation and nλ relates to the generalized differentiation which can be associated to spacial differentiation

A guided tour of new tempered distributions

Laurent Schwartz, the principle architect of distribution theory, presented the impossibility of extending a form of multiplication to distribution theory. There have been many varieties of partial solutions to this problem. Some of the solutions contain heuristic computations done by physicists in quantum field theory. A recent strategy developed by J. Colombeau culminates with multiplication and integration theory for distributions. This paper develops this theory in the spirit of a sequence approach, much like fundamental sequences are to distributions. However, in the new tempered distribution theory the sequences can be noncountable. T. Todorov developed these techniques for new distributions. However, since so many applications require Fourier analysis, the new tempered distributions provide a natural setting for physics and signal analysis. The paper illustrates the product of two Dirac delta functionals,δ(x)δ(x). Other nonregular distributional products can also be computed in the same manner. The paper culminates with a new application of annihilation and creation operators in quantum field theory.

Elementary analysis through examples and exercises

Preface. 1. Real numbers. 2. Functions. 3. Sequences. 4. Limits of functions. 5. Continuity. 6. Derivatives. 7. Graphs of functions. Bibliography. Index.

Fourier transforms in generalized Fock spaces

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),where ϕ0∈C and ϕq∈L2(R3q), q≥1. We will replace the ϕq, q≥1 with q-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter, s, which sweeps out a scale of generalized Fock spaces.

Multiplication and mixed differentiation in generalized Fock spaces

The classical distribution theory founded by Laurent Schwartz [26] has an intrinsic multiplication problem. He indicates this in his paper [27]. This problem is clearly demonstrated by selecting two particular distributions and attempting to define a multiplication that enjoys commutative, associative, and unit identity properties. This process leads to the contradiction, 1 = eP I [lo]. A new generalized function theory developed by J. Colombeau [4-91 provides a solution to the multiplication problem and also develops a general integration theory. T. Todorov reexamines the new generalized function theory in the spirit of non-denumerable sequences termed ultrapowers [29]. The L. Schwartz distribution theory was reexamined in the spirit of fundamental sequences [ 16,333. However, there are new considerations in the theory of new generalized function theory requiring an algebraic notion of a particular ideal contained in a specified test function space. A brief review of these constructions can be found in Section 7 and a comprehensive review can be found in Ref. [25]. A principle application for our development will be modeling a physical system consisting of an “infinite” number of particles. A system of n-particles termed bosoms or fermions has previously been modeled by n-dimensional symmetric or antisymmetric functions. When this system is enlarged to an infinite number of particles, the model for the state space vector has the form,

Stieltjes transforms on new generalized functions

We introduce a Stieltjes transform on the equivalence classes of a new generalized function which has been successfully developed by Colombeau. Subsets of rapid descent test functions, 𝒮(ℝn), as well as tempered distributions, 𝒮′(ℝn), are used to preserve Fourier analysis techniques.