Real and Complex Analysis

Foundations Of Time Frequency Analysis

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

As for the fractional transforms, to date only fractional Fourier transform (FrFT) has been applicable to color images in a holistic manner by using the quaternion algebra, yet, discrete fractional random transform (DFRNT) with the useful intrinsic randomness is still to be explored. This paper first defines quaternion DFRNT (QDFRNT) which generalizes DFRNT to efficiently process quaternion signals, and then applies QDFRNT to color image adaptive watermarking. For the QDFRNT, this paper also derives the relationship between QDFRNT of a quaternion signal and DFRNT of four components for this signal to efficiently compute QDFRNT. For the color image adaptive watermarking based on QDFRNT and SVM, in order to efficiently utilize the color information in the adaptive process, this paper also exploits the human vision system’s (HVS) masking properties of texture, edge and color tone directly from the color host image to adaptively adjust the watermark strength for each block. In addition, the constraints in watermark embedding are discussed to preserve the watermarking energy. Experimental results show that: (a) the proposed efficient computation method takes only half the computational time of the direct method; (b) the comparison of five color models (RGB, YUV, YIQ, CIEL*a*b* and YCbCr) shows that the proposed QDFRNT-based watermarking scheme using YCbCr color model has the overall best performance and can achieve a good balance between invisibility and robustness to the Checkmark attacks; (c) The proposed scheme is superior to the existing schemes respectively using DCT, DFRNT, discrete quaternion Fourier transform (DQFT), discrete fractional quaternion Fourier transform (DFrQFT), and quaternion radial moments (QRMs). Moreover, the fractional order and the random matrix of QDFRNT enhance the security of the proposed scheme.

Quaternion discrete fractional random transform for color image adaptive watermarking

In this study, by using the quaternion algebra, multiple-parameter fractional quaternion Fourier transform (MPFrQFT) is proposed to generalise the conventional multiple-parameter fractional Fourier transform (MPFrFT) to quaternion signal processing in a holistic manner. First, the new transform MPFrQFT and its inverse transform are defined. An efficient discrete implementation method of MPFrQFT is then proposed, in which the relationship between MPFrQFT and MPFrFT of four components is utilised for a quaternion signal. Finally, a new colour image encryption algorithm based on the proposed MPFrQFT and the double random phase encoding technique is proposed to evaluate the performance of the proposed MPFrQFT. Experimental results demonstrate that: (i) the computational time of the proposed implementation method is almost a half of the direct method's time; (ii) the proposed MPFrQFT-based encryption algorithm has an overall better performance than eight compared algorithms in security test and robustness test: it is more secure than the compared frequency-based algorithms due to the larger key space and the more sensitive key `transform orders'; it is also more robust than the compared spatial-domain algorithms.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-ipr.2018.5440

Multiple-parameter fractional quaternion Fourier transform and its application in colour image encryption

Quaternionic one-dimensional fractional Fourier transform

Quaternionic curvelet transform

The coupled fractional Fourier transform is a two-dimensional fractional Fourier transform that depends on two angles that are coupled in such a way that the transform parameters are and It generalizes the two-dimensional Fourier transform and it serves as useful tool in some applications in optics and signal processing. In this article we derive new properties of the transform, such as its additive property. We then extend some of them to and show that the transform is a unitary operator on

On the extension of the coupled fractional Fourier transform and its properties

Short time coupled fractional fourier transform and the uncertainty principle

We extend the special affine Fourier transform in the context of quaternion valued functions and study its properties including an uncertainty principle. The same transform is studied on a suitably constructed Boehmian space.

Rajakumar Roopkumar

Papers

Quaternionic one-dimensional fractional Fourier transform

Quaternionic curvelet transform

On the extension of the coupled fractional Fourier transform and its properties

Short time coupled fractional fourier transform and the uncertainty principle

One-Dimensional Quaternionic Special Affine Fourier Transform