# Functions and Analysis

## 1306 Submissions

 viXra:1306.0228 [pdf] submitted on 2013-06-28 13:11:44

### A Mathematical Conjecture on the Wavefunctions of Quantum Mechanics

Authors: HaengJin Choe

The uncertainty principle is one of the fundamental principles of quantum mechanics. While studying quantum mechanics recently, the author made an exciting mathematical discovery about the product of two expectation values. The author explains the discovery.
Category: Functions and Analysis

 viXra:1306.0133 [pdf] submitted on 2013-06-17 05:21:04

### Tutorial on Fourier Transformations and Wavelet Transformations in Cliord Geometric Algebra

Authors: Eckhard Hitzer
Comments: 45 Pages. 3 tables. In K. Tachibana (ed.) Tutorial on Fourier Transf. and Wavelet Transf. in Clifford Geometric Algebra, Lect. notes of the Int. Workshop for “Computational Science with Geometric Algebra” (FCSGA2007), Nagoya Univ., JP, Feb. 2007, pp. 65-87 (2007).

First, the basic concept multivector functions and their vector derivative in geometric algebra (GA) is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on GA multivector-valued functions (f : R^3 -> Cl(3,0)). Third, we show a set of important properties of the Clifford Fourier transform (CFT) on Cl(3,0) such as dierentiation properties, and the Plancherel theorem. We round o the treatment of the CFT (at the end of this tutorial) by applying the Clifford Fourier transform properties for proving an uncertainty principle for Cl(3,0) multivector functions. For wavelets in GA it is shown how continuous Clifford Cl(3,0)- valued admissible wavelets can be constructed using the similitude group SIM(3), a subgroup of the ane group of R^3. We express the admissibility condition in terms of the CFT and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform of multivector functions. We explain (at the end of this tutorial) a generalized Clifford wavelet uncertainty principle. For scalar admissibility constant it sets bounds of accuracy in multivector wavelet signal and image processing. As concrete example we introduce multivector Clifford Gabor wavelets, and describe important properties such as the Clifford Gabor transform isometry, a reconstruction formula, and (at the end of this tutorial) an uncertainty principle for Clifford Gabor wavelets. Keywords: vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle, similitude group, geometric algebra wavelet transform, geometric algebra Gabor wavelets.
Category: Functions and Analysis

 viXra:1306.0130 [pdf] submitted on 2013-06-17 01:29:18

### The Clifford Fourier Transform in Real Clifford Algebras

Authors: Eckhard Hitzer
Comments: 21 Pages. 2 figures, 1 table. First published: Proc. of 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany, 04–06 July 2012.

We use the recent comprehensive research [17, 19] on the manifolds of square roots of -1 in real Clifford’s geometric algebras Cl(p,q) in order to construct the Clifford Fourier transform. Basically in the kernel of the complex Fourier transform the imaginary unit j in C (complex numbers) is replaced by a square root of -1 in Cl(p,q). The Clifford Fourier transform (CFT) thus obtained generalizes previously known and applied CFTs [9, 13, 14], which replaced j in C only by blades (usually pseudoscalars) squaring to -1. A major advantage of real Clifford algebra CFTs is their completely real geometric interpretation. We study (left and right) linearity of the CFT for constant multivector coefficients in Cl(p,q), translation (x-shift) and modulation (w-shift) properties, and signal dilations. We show an inversion theorem. We establish the CFT of vector differentials, partial derivatives, vector derivatives and spatial moments of the signal. We also derive Plancherel and Parseval identities as well as a general convolution theorem. Keywords: Clifford Fourier transform, Clifford algebra, signal processing, square roots of -1.
Category: Functions and Analysis

 viXra:1306.0127 [pdf] submitted on 2013-06-17 01:59:58

### Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions N = 2 (Mod 4) and N = 3 (Mod 4)

Authors: Eckhard Hitzer, Bahri Mawardi
Comments: 24 Pages. 2 tables. Adv. App. Cliff. Alg. Vol. 18, S3,4, pp. 715-736 (2008). DOI: 10.1007/s00006-008-0098-3.

First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. Second, we dene a generalized real Fourier transform on Clifford multivector-valued functions ( f : R^n -> Cl(n,0), n = 2,3 (mod 4) ). Third, we show a set of important properties of the Clifford Fourier transform on Cl(n,0), n = 2,3 (mod 4) such as dierentiation properties, and the Plancherel theorem, independent of special commutation properties. Fourth, we develop and utilize commutation properties for giving explicit formulas for f x^m; f Nabla^m and for the Clifford convolution. Finally, we apply Clifford Fourier transform properties for proving an uncertainty principle for Cl(n,0), n = 2,3 (mod 4) multivector functions. Keywords: Vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle.
Category: Functions and Analysis

 viXra:1306.0126 [pdf] submitted on 2013-06-17 02:09:49

### Uncertainty Principle for Clifford Geometric Algebras Cl(n,0), N = 3 (Mod 4) Based on Clifford Fourier Transform

Authors: Eckhard Hitzer, Bahri Mawardi
Comments: 10 Pages. 1 table. In T. Qian, M.I. Vai, X. Yusheng (eds.), Wavelet Analysis and Applications, Springer (SCI) Book Series Applied and Numerical Harmonic Analysis, Springer, pp. 45-54 (2006). DOI: 10.1007/978-3-7643-7778-6_6.

First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. Second, we define a generalized real Fourier transform on Clifford multivector-valued functions (f : Rn -> Cl(n,0), n = 3 (mod 4)). Third, we introduce a set of important properties of the Clifford Fourier transform on Cl(n,0), n = 3 (mod 4) such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving a directional uncertainty principle for Cl(n,0), n = 3 (mod 4) multivector functions. Keywords. Vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle. Mathematics Subject Classication (2000). Primary 15A66; Secondary 43A32.
Category: Functions and Analysis

 viXra:1306.0124 [pdf] submitted on 2013-06-17 02:53:25

### Basic Multivector Calculus

Authors: Eckhard Hitzer
Comments: 6 Pages. Proc. of 18th Intelligent Systems Symposium (FAN 2008), 23-24 Oct. 2008, Hiroshima, Japan, pp. 185 – 190 (2008).

We begin with introducing the generalization of real, complex, and quaternion numbers to hypercomplex numbers, also known as Clifford numbers, or multivectors of geometric algebra. Multivectors encode everything from vectors, rotations, scaling transformations, improper transformations (reflections, inversions), geometric objects (like lines and spheres), spinors, and tensors, and the like. Multivector calculus allows to define functions mapping multivectors to multivectors, differentiation, integration, function norms, multivector Fourier transformations and wavelet transformations, filtering, windowing, etc. We give a basic introduction into this general mathematical language, which has fascinating applications in physics, engineering, and computer science.
Category: Functions and Analysis

 viXra:1306.0122 [pdf] submitted on 2013-06-17 03:06:17

### Foundations of Multidimensional Wavelet Theory: The Quaternion Fourier Transform and its Generalizations

Authors: Eckhard Hitzer
Comments: 3 Pages. E. Hitzer, Foundations of Multidimensional Wavelet Theory: The Quaternion Fourier Transf. and its Generalizations, Preprints of Meeting of the JSIAM, ISSN: 1345-3378, Tsukuba Univ., 16-18 Sep. 2006, Tsukuba, Japan, pp. 66,67.

Keywords: Multidimensional Wavelets, Quaternion Fourier Transform, Clifford geometric algebra
Category: Functions and Analysis

 viXra:1306.0117 [pdf] submitted on 2013-06-17 03:56:01

### Geometric Calculus – Engineering Mathematics for the 21st Century

Authors: Eckhard Hitzer
Comments: 12 Pages. 13 figures. Mem. Fac. Eng. Fukui Univ. 50(1), pp. 127-137 (2002).

This paper treats important questions at the interface of mathematics and the engineering sciences. It starts off with a quick quotation tour through 2300 years of mathematical history. At the beginning of the 21st century, technology has developed beyond every expectation. But do we also learn and practice an adequately modern form of mathematics? The paper argues that this role is very likely to be played by universal geometric calculus. The fundamental geometric product of vectors is introduced. This gives a quick-and-easy description of rotations as well as the ultimate geometric interpretation of the famous quaternions of Sir W.R. Hamilton. Then follows a one page review of the historical roots of geometric calculus. In order to exemplify the role of geometric calculus for the engineering sciences three representative examples are looked at in some detail: elasticity, image geometry and pose estimation. Next a current snapshot survey of geometric calculus software is provided. Finally the value of geometric calculus for teaching, research and development is commented.
Category: Functions and Analysis

 viXra:1306.0116 [pdf] submitted on 2013-06-17 04:00:42

### Vector Differential Calculus

Authors: Eckhard Hitzer
Comments: 17 Pages. Mem. Fac. Eng. Fukui Univ. 50(1), pp. 109-125 (2002).

This paper treats the fundamentals of the vector differential calculus part of universal geometric calculus. Geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. In order to make the treatment self-contained, I first compile all important geometric algebra relationships, which are necessary for vector differential calculus. Then differentiation by vectors is introduced and a host of major vector differential and vector derivative relationships is proven explicitly in a very elementary step by step approach. The paper is thus intended to serve as reference material, giving details, which are usually skipped in more advanced discussions of the subject matter. Keywords: Geometric Calculus, Geometric Algebra, Clifford Algebra, Vector Derivative, Vector Differential Calculus
Category: Functions and Analysis

 viXra:1306.0114 [pdf] submitted on 2013-06-17 04:13:56

### Geometric Calculus for Engineers

Authors: Eckhard Hitzer
Comments: 8 Pages. 7 figures. Proc. of the Pukyong National University - Fukui University International Symposium 2001 for Promotion of Research Cooperation, Pukyong National University, Busan, Korea, pp. 59-66 (2001).

This paper treats important questions at the interface of mathematics and the engineering sciences. It starts off with a quick quotation tour through 2300 years of mathematical history. At the beginning of the 21st century, technology has developed beyond every expectation. But do we also learn and practice an adequately modern form of mathematics? The paper argues that this role is very likely to be played by (universal) geometric calculus. The fundamental geometric product of vectors is introduced. This gives a quick-and-easy description of rotations as well as the ultimate geometric interpretation of the famous quaternions of Sir W.R. Hamilton. Then follows a one page review of the historical roots of geometric calculus. In order to exemplify the role geometric calculus for the engineering sciences three representative examples are looked at in some detail: elasticity, image geometry and pose estimation. Finally the value of geometric calculus for teaching, research and development and its worldwide impact are commented.
Category: Functions and Analysis

 viXra:1306.0096 [pdf] submitted on 2013-06-14 03:17:09

### Windowed Fourier Transform of Two-Dimensional Quaternionic Signals

Authors: B. Mawardi, E. Hitzer, R. Ashino, R. Vaillancourt
Comments: 20 Pages. Appl. Math. and Computation, 216, Iss. 8, pp. 2366-2379, 15 June 2010. 6 figures, 1 table.

In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral representation of the quaternionic Fourier transform (QFT), we derive several important properties such as reconstruction formula, reproducing kernel, isometry, and orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic Gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic Gabor basis. We apply the QWFT properties and the (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT. Finally, we briefly introduce an application of the QWFT to a linear time-varying system. Keywords: quaternionic Fourier transform, quaternionic windowed Fourier transform, signal processing, Heisenberg type uncertainty principle
Category: Functions and Analysis

 viXra:1306.0095 [pdf] submitted on 2013-06-14 03:21:42

### Clifford Algebra Cl(3,0)-valued Wavelets and Uncertainty Inequality for Clifford Gabor Wavelet Transformation

Authors: Mawardi Bahri, Eckhard Hitzer
Comments: 2 Pages. Preprints of Meeting of the Japan Society for Industrial and Applied Mathematics, ISSN: 1345-3378, Tsukuba University, 16-18 Sep. 2006, Tsukuba, Japan, pp. 64,65.

The purpose of this paper is to construct Clifford algebra Cl(3,0)-valued wavelets using the similitude group SIM(3) and then give a detailed explanation of their properties using the Clifford Fourier transform. Our approach can generalize complex Gabor wavelets to multivectors called Clifford Gabor wavelets. Finally, we describe some of their important properties which we use to establish a new uncertainty principle for the Clifford Gabor wavelet transform.
Category: Functions and Analysis

 viXra:1306.0094 [pdf] submitted on 2013-06-14 03:35:04

### Clifford Algebra Cl(3,0)-valued Wavelet Transformation, Clifford Wavelet Uncertainty Inequality and Clifford Gabor Wavelets

Authors: Mawardi Bahri, Eckhard Hitzer
Comments: 23 Pages. International Journal of Wavelets, Multiresolution and Information Processing, 5(6), pp. 997-1019 (2007). DOI: 10.1142/S0219691307002166, 2 tables.

In this paper, it is shown how continuous Clifford Cl(3,0)-valued admissible wavelets can be constructed using the similitude group SIM(3), a subgroup of the affine group of R^3. We express the admissibility condition in terms of a Cl(3,0) Clifford Fourier transform and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform of multivector functions. We invent a generalized Clifford wavelet uncertainty principle. For scalar admissibility constant it sets bounds of accuracy in multivector wavelet signal and image processing. As concrete example we introduce multivector Clifford Gabor wavelets, and describe important properties such as the Clifford Gabor transform isometry, a reconstruction formula, and an uncertainty principle for Clifford Gabor wavelets. Keywords: Similitude group, Clifford Fourier transform, Clifford wavelet transform, Clifford Gabor wavelets, uncertainty principle.
Category: Functions and Analysis

 viXra:1306.0092 [pdf] submitted on 2013-06-14 04:25:26

### Two-Dimensional Clifford Windowed Fourier Transform

Authors: Mawardi Bahri, Eckhard Hitzer, Sriwulan Adji
Comments: 15 Pages. in G. Scheuermann, E. Bayro-Corrochano (eds.), Geometric Algebra Computing, Springer, New York, 2010, pp. 93-106. 4 figures, 1 table.

Recently several generalizations to higher dimension of the classical Fourier transform (FT) using Clifford geometric algebra have been introduced, including the two-dimensional (2D) Clifford Fourier transform (CFT). Based on the 2D CFT, we establish the two-dimensional Clifford windowed Fourier transform (CWFT). Using the spectral representation of the CFT, we derive several important properties such as shift, modulation, a reproducing kernel, isometry and an orthogonality relation. Finally, we discuss examples of the CWFT and compare the CFT and the CWFT.
Category: Functions and Analysis

 viXra:1306.0091 [pdf] submitted on 2013-06-14 04:35:57

### An Uncertainty Principle for Quaternion Fourier Transform

Authors: Mawardi Bahri, Eckhard Hitzer, Akihisa Hayashi, Ryuichi Ashino
Comments: 20 Pages. Computer & Mathematics with Applications, 56, pp. 2398-2410 (2008). DOI: 10.1016/j.camwa.2008.05.032, 3 figures, 1 table.

We review the quaternionic Fourier transform (QFT). Using the properties of the QFT we establish an uncertainty principle for the right-sided QFT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternion signal minimizes the uncertainty. Key words: Quaternion algebra, Quaternionic Fourier transform, Uncertainty principle, Gaussian quaternion signal, Hypercomplex functions Math. Subj. Class.: 30G35, 42B10, 94A12, 11R52
Category: Functions and Analysis

 viXra:1306.0089 [pdf] submitted on 2013-06-14 04:44:58

### Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl(3,0)

Authors: Bahri Mawardi, Eckhard Hitzer
Comments: 23 Pages. Advances in Applied Clifford Algebras, 16(1), pp. 41-61 (2006). DOI 10.1007/s00006-006-0003-x , 3 tables.

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions (f: R^3 -> Cl(3,0)). Third, we show a set of important properties of the Clifford Fourier transform on Cl(3,0) such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl(3,0) multivector functions. Keywords: vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle.
Category: Functions and Analysis