Integral transforms (Laplace, Fourier and Mellin) are introduced with their properties, the relationship between these transforms was discussed and some important applications in physics and engineering were given. ااااااا دقل مت ضارعتسإ ةساردو ل ةيلماكتلا تليوحتلا لك ، سلبل تلوحت نم روف ي ر نيليمو عم ةشقانم كلذكو ،اهنم لك صاوخ و صئاصخ ةقلعلا ةشقانم مت هذه نيب طبرلاو و ،تليوحتلا مت ميدقت تاقيبطتلا ضعب تليوحتلا هذهل ةمهملا يف تلاجم ءايزيفلا ةسدنهلاو.

Integral transforms and their applications

The Convolution Transform. By Hirschmann & Widder . Pp. x + 268. 45s. 1955. (Princeton University Press. London : Cumberlege)

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.)

Operational Calculus By Prof Jan Mikusinski (International Series of Monographs on Pure and Applied Mathematics, Vol 8) Pp 495 (London and New York: Pergamon Press; Warszawa: Panstwowe Wydawnictwo Naukowe, 1959) 100s net

The Convolution Transform

This chapter includes proofs of the existence and uniqueness of the solutions of first order systems of q-difference equations in a neighborhood of a point a, \(a \geq 0\). Then, as applications of the main results, we study linear q-difference equations as well as the q-type Wronskian. These results are mainly based on (Mansour, q-Difference Equations, Master’s thesis, Faculty of Science, Cairo University, Giza, Egypt, 2001). This chapter also includes a section on the asymptotics of zeros of some q-functions.

q -Difference Equations

A q-analogue of the Mellin transform is introduced by using a standard method of q-calculus involving the q-Jackson integral. In this paper, we study some of its properties coinciding with the corresponding classical ones when q tends to 1. In addition to several examples given, we establish the q-inversion formula, the q-analogue of the convolution product, and the q-extension of the known Titchmarsh theorem. Finally, we prove the q-Mellin summation formula related to some q-zeta function.

The Mellin Transform in Quantum Calculus

The q-jα Bessel Function

We generalize the theory of the heat polynomials introduced by P. V. Rosenbloom and D. V. Widder for a more general class of singular differential operator on (0, ∞). The heat polynomials associated with the Bessel operator and studied by D. T. Haimo appear as a particular case in this paper. In the special cases of second derivative and Bessel operators the heat polynomials are in fact polynomials inx andt, however, this property does not hold in general.

Heat “polynomials” for a singular differential operator on (0, ∞)

In this paper we introduce the q−Bessel Fourier transform, the q−Bessel translation operator and the q−convolution product. We prove that the q−heat semigroup is contractive and we establish the q−analogue of Babenko inequalities associated to the q−Bessel Fourier transform. With applications and finally we enunciate a q−Bessel version of the central limit theorem.

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INEQUALITIES IN q FOURIER ANALYSIS

Using a new formulation of Graf’s addition formula related to the third Jackson q-Bessel function, we study the positivity of the generalized q-translation operator associated with the q-Hankel transform.

/pdf/positivity-of-the-generalized-translation-associated-with-2rbz2ireop.pdf

Ahmed Fitouhi

Papers

The Mellin Transform in Quantum Calculus

The q-jα Bessel Function

Heat “polynomials” for a singular differential operator on (0, ∞)

INEQUALITIES IN q FOURIER ANALYSIS

Positivity of the Generalized Translation Associated with the q-Hankel Transform