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Axioms, Volume 2, Issue 2 (June 2013), Pages 67-285

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Research

Open AccessArticle Mollification Based on Wavelets
Axioms 2013, 2(2), 67-84; doi:10.3390/axioms2020067
Received: 9 January 2013 / Revised: 11 March 2013 / Accepted: 19 March 2013 / Published: 25 March 2013
Cited by 1 | PDF Full-text (626 KB) | HTML Full-text | XML Full-text
Abstract
The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case
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The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case when the fractional derivative of a function is calculated. Noting that the estimations are applicable even when the orthogonality of the wavelets is not satisfied, we study mollifications using unorthogonalized wavelets, as well as those using orthogonal wavelets. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle On the q-Analogues of Srivastava’s Triple Hypergeometric Functions
Axioms 2013, 2(2), 85-99; doi:10.3390/axioms2020085
Received: 28 February 2013 / Revised: 22 March 2013 / Accepted: 3 April 2013 / Published: 11 April 2013
Cited by 1 | PDF Full-text (207 KB) | HTML Full-text | XML Full-text
Abstract
We find Euler integral formulas, summation and reduction formulas for q-analogues of Srivastava’s three triple hypergeometric functions. The proofs use q-analogues of Picard’s integral formula for the first Appell function, a summation formula for the first Appell function based on the Bayley–Daum formula,
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We find Euler integral formulas, summation and reduction formulas for q-analogues of Srivastava’s three triple hypergeometric functions. The proofs use q-analogues of Picard’s integral formula for the first Appell function, a summation formula for the first Appell function based on the Bayley–Daum formula, and a general triple series reduction formula of Karlsson. Many of the formulas are purely formal, since it is difficult to find convergence regions for these functions of several complex variables. We use the Ward q-addition to describe the known convergence regions of q-Appell and q-Lauricella functions. Full article
Open AccessArticle Divergence-Free Multiwavelets on the Half Plane
Axioms 2013, 2(2), 100-121; doi:10.3390/axioms2020100
Received: 24 December 2012 / Revised: 16 March 2013 / Accepted: 18 March 2013 / Published: 11 April 2013
Cited by 2 | PDF Full-text (528 KB) | HTML Full-text | XML Full-text
Abstract
We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal
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We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R2 +. Such wavelets are suitable to study the Navier–Stokes equations on a half plane when imposing a Navier boundary condition. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Construction of Multiwavelets on an Interval
Axioms 2013, 2(2), 122-141; doi:10.3390/axioms2020122
Received: 5 February 2013 / Revised: 21 March 2013 / Accepted: 26 March 2013 / Published: 17 April 2013
Cited by 1 | PDF Full-text (260 KB) | HTML Full-text | XML Full-text
Abstract
Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article
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Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article we show how an algorithm of Madych for scalar wavelets can be generalized to multiwavelets, given an extra assumption. We then develop a new algorithm that does not require this additional condition. Finally, we apply results from a previous paper to resolve the non-uniqueness of the algorithm by imposing regularity conditions (including approximation orders) on the boundary functions. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations
Axioms 2013, 2(2), 142-181; doi:10.3390/axioms2020142
Received: 28 February 2013 / Accepted: 8 April 2013 / Published: 23 April 2013
Cited by 1 | PDF Full-text (907 KB) | HTML Full-text | XML Full-text
Abstract
This paper describes and tests a wavelet-based implicit numerical method for solving partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time-discretization into multiple smaller
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This paper describes and tests a wavelet-based implicit numerical method for solving partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time-discretization into multiple smaller systems that can be solved sequentially. Included is a test on a basic non-linear problem, with both the results of the test, and the time required to calculate them, compared with control results based on a single system with fine resolution. The method is then tested on a non-trivial problem, its computational time and accuracy checked against control results. In both tests, it was found that the method requires less computational expense than the control. Furthermore, the method showed convergence towards the fine resolution control results. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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Open AccessArticle Time Scale Analysis of Interest Rate Spreads and Output Using Wavelets
Axioms 2013, 2(2), 182-207; doi:10.3390/axioms2020182
Received: 16 February 2013 / Revised: 18 March 2013 / Accepted: 28 March 2013 / Published: 23 April 2013
Cited by 1 | PDF Full-text (1336 KB) | HTML Full-text | XML Full-text
Abstract
This paper adds to the literature on the information content of different spreads for real activity by explicitly taking into account the time scale relationship between a variety of monetary and financial indicators (real interest rate, term and credit spreads) and output growth.
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This paper adds to the literature on the information content of different spreads for real activity by explicitly taking into account the time scale relationship between a variety of monetary and financial indicators (real interest rate, term and credit spreads) and output growth. By means of wavelet-based exploratory data analysis we obtain richer results relative to the aggregate analysis by identifying the dominant scales of variation in the data and the scales and location at which structural breaks have occurred. Moreover, using the “double residuals” regression analysis on a scale-by-scale basis, we find that changes in the spread in several markets have different information content for output at different time frames. This is consistent with the idea that allowing for different time scales of variation in the data can provide a fruitful understanding of the complex dynamics of economic relationships between variables with non-stationary or transient components, certainly richer than those obtained using standard time domain methods. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Using the Choquet Integral in the Fuzzy Reasoning Method of Fuzzy Rule-Based Classification Systems
Axioms 2013, 2(2), 208-223; doi:10.3390/axioms2020208
Received: 25 February 2013 / Revised: 21 March 2013 / Accepted: 3 April 2013 / Published: 23 April 2013
Cited by 6 | PDF Full-text (252 KB) | HTML Full-text | XML Full-text
Abstract
In this paper we present a new fuzzy reasoning method in which the Choquet integral is used as aggregation function. In this manner, we can take into account the interaction among the rules of the system. For this reason, we consider several fuzzy
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In this paper we present a new fuzzy reasoning method in which the Choquet integral is used as aggregation function. In this manner, we can take into account the interaction among the rules of the system. For this reason, we consider several fuzzy measures, since it is a key point on the subsequent success of the Choquet integral, and we apply the new method with the same fuzzy measure for all the classes. However, the relationship among the set of rules of each class can be different and therefore the best fuzzy measure can change depending on the class. Consequently, we propose a learning method by means of a genetic algorithm in which the most suitable fuzzy measure for each class is computed. From the obtained results it is shown that our new proposal allows the performance of the classical fuzzy reasoning methods of the winning rule and additive combination to be enhanced whenever the fuzzy measure is appropriate for the tackled problem. Full article
(This article belongs to the Special Issue Axiomatic Approach to Monotone Measures and Integrals)
Open AccessArticle Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces
Axioms 2013, 2(2), 224-270; doi:10.3390/axioms2020224
Received: 4 March 2013 / Revised: 12 May 2013 / Accepted: 14 May 2013 / Published: 23 May 2013
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Abstract
We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions
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We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hölder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards
Axioms 2013, 2(2), 271-285; doi:10.3390/axioms2020271
Received: 24 April 2013 / Revised: 17 May 2013 / Accepted: 22 May 2013 / Published: 18 June 2013
Cited by 2 | PDF Full-text (607 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
Wavelet analysis is known to be a good option for change detection in many contexts. Detecting changes in solution volumes that are measured with both additive and relative error is an important aspect of safeguards for facilities that process special nuclear material. This
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Wavelet analysis is known to be a good option for change detection in many contexts. Detecting changes in solution volumes that are measured with both additive and relative error is an important aspect of safeguards for facilities that process special nuclear material. This paper qualitatively compares wavelet-based change detection to a lag-one differencing option using realistic simulated solution volume data for which the true change points are known. We then show quantitatively that Haar wavelet-based change detection is effective for finding the approximate location of each change point, and that a simple piecewise linear optimization step is effective to refine the initial wavelet-based change point estimate. Full article
(This article belongs to the Special Issue Wavelets and Applications)

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