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

**Authors:** HaengJin Choe

**Comments:** 4 Pages.

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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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