Complex analysis with applications

This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling this text to serve well in courses for students in engineering or applied sciences. The overall aim in designing this text is to accommodate students of different mathematical backgrounds and to achieve a balance between presentations of rigorous mathematical proofs and applications. The text is adapted to enable maximum flexibility to instructors and to students who may also choose to progress through the material outside of coursework. Detailed examples may be covered in one course, giving the instructor the option to choose those that are best suited for discussion. Examples showcase a variety of problems with completely worked out solutions, assisting students in working through the exercises. The numerous exercises vary in difficulty from simple applications of formulas to more advanced project-type problems. Detailed hints accompany the more challenging problems. Multi-part exercises may be assigned to individual students, to groups as projects, or serve as further illustrations for the instructor. Widely used graphics clarify both concrete and abstract concepts, helping students visualize the proofs of many results. Freely accessible solutions to every-other-odd exercise are posted to the book’s Springer website. Additional solutions for instructors’ use may be obtained by contacting the authors directly

On a weak type (1

1) inequality for a maximal conjugate functio

On a weak type (1,1) inequality for a maximal conjugate function

In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of $H^p$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus

Hahn&apos;s embedding theorem for orders and harmonic analysis on groups with ordered duals

Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on L p (G), 1 ≤ p &lt; ∞, as was done by Helson [7]. We will show how to use Hahn’s Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in L p (G). This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series.

Analytic measures and Bochner measurability

Many authors have made great strides in extending the celebrated F. and M. Riesz Theorem to various abstract settings. Most notably, we have, in chronological order, the work of Bochner [1], Helson and Lowdenslager [7], de Leeuw and Glicksberg [4], and Forelli [6]. These formidable papers build on each other’s ideas and provid

Hardy martingales and Jensen’s Inequality

Hardy martingales were introduced by Garling and used to study analytic functions on the N-dimensional torus T N, where analyticity is defined using a lexicographic order on the dual group Z N. We show how, by using basic properties of orders on Z N, we can apply Garling’s method in the study of analytic functions on an arbitrary compact abelian group with an arbitrary order on its dual group. We illustrate our approach by giving a new and simple proof of a famous generalized Jensen’s Inequality due to Helson and Lowdenslager [5].