There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

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.)

Stochastic processes and the Wiener integral

In this paper we develop an Lp Fourier-Feynman theory for a class of functionals on Wiener space of the form F(x) = f(J0 axdx, ... , /0 a„dx). We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.

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Analytic Fourier-Feynman TransformsAnd Convolution

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Conditional Function Space Integrals with Applications

In this paper we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We establish a translation theorem and use it to express the generalized Feynman integral of the first variation of a functional F in terms of the generalized Feynman integral of F multiplied by a linear factor. We establish some integration by parts formulas for generalized Feynman integrals and transforms. We also find the generalized Fourier-Feynman transform of a functional F belonging to a Banach algebra &#x1d4ae;(L 2 a, b [0, T]) after it has been multiplied by n linear factors; none of these linear factors belong to &#x1d4ae;(L 2 a, b [0, T]). Finally we established some new generalized Feynman integration formulas.

Generalized Fourier-Feynman transforms and a first variation on function space

In this paper we use generalized Fourier-Hermite functionals to obtain a complete orthonormal set in L2(Ca,b[0,T]) where Ca,b[0,T] is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in L2(Ca,b[0,T]) has an integral transform ℱγ,βF also belonging to L2(Ca,b[0,T]).

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Integral Transforms of Functionals in L2(Ca,b[0,T])

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form F(x) = f( ,..., ) where denotes the Paley-Wiener-Zygmund stochastic integral ∫ T 0 α(t)dx(t).

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Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

In this paper, we establish several very basic formulas relating convolution products, integral transforms and inverse integral transforms for functionals in L 2(C 0[0, T]). The expansion of functionals in L 2(C 0[0, T]) in terms of Fourier–Hermite functionals plays a key role.

Seung Jun Chang

Papers

Conditional Function Space Integrals with Applications

Generalized Fourier-Feynman transforms and a first variation on function space

Integral Transforms of Functionals in L2(Ca,b[0,T])

Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

Convolution products, integral transforms and inverse integral transforms of functionals in L 2(C 0[0, T])