TAILIEUCHUNG - Wavelets trong Electromagnetics và mô hình thiết bị P1

Notations and Mathematical Preliminaries NOTATIONS AND ABBREVIATIONS The notations and abbreviations used in the book are summarized here for ease of reference. D (α) f = f α (t) := d f α (t)/dt α f¯—complex conjugate of f ∞ fˆ := −∞ f (t)e−iωt dt, Fourier transform of f (t) ∞ 1 f (t) := 2π −∞ fˆ(ω)eiωt dω, inverse Fourier transform of fˆ(ω) f —norm of a function f ∗ g—convolution f, h := f (t)h(t) dt, inner product f n = O(n)-order of n, ∃C such that f n ≤ Cn C—complex N —nonnegative integers R—real number. | Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright 2003 John Wiley Sons Inc. ISBN 0-471-41901-X CHAPTER ONE Notations and Mathematical Preliminaries NOTATIONS AND ABBREVIATIONS The notations and abbreviations used in the book are summarized here for ease of reference. D a f f a t df a t dt f complex conjugate of f f - f t e lmt dt Fourier transform of f t f t 27 - f CAe1 mt do inverse Fourier transform of f 01 f norm of a function f g convolution f h f f t h t dt inner product fn O n -order of n 3C such that fn Cn C complex N nonnegative integers R real number Rn real numbers of size n Z integers Z positive integers L2 R functional space consisting finite energy functions f t 2 dt Lp R function space that f f t p dt 12 Z finite energy series X v an I2 f 2 set Hs Q Ws 2 -Sobolev space equipped with inner product of U v s 2 T. a s DauDavd 1 2 NOTATIONS AND MATHEMATICAL PRELIMINARIES V W direct sum V W tensor product V f gradient H E vector fields V x H curl V E divergence a J largest integer m a 8m n Kronecker delta 8 t Dirac delta X a b characteristic function which is 1 in a b and zero outside end of proof 3 exist V any iff if and only if . almost everywhere . direct current . orthonormal . otherwise MATHEMATICAL PRELIMINARIES This chapter is arranged here to familiarize the reader with the mathematical notation definitions and theorems that are used in wavelet literature and in this book. Important mathematical concepts are briefly reviewed. In most cases no proof is given. For more detailed discussions or in depth studies readers are referred to the corresponding references 1-5 . Readers are suggested to skip this chapter in their first reading. They may then return to the relevant sections of this chapter if unfamiliar mathematical concepts present themselves during the course of the book. Functions and Integration A function f t is called integrable if f f t dt to J to and we say that f e L 1 R . Two .

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