Fourier Analysis STUST傅里叶分析sust.docVIP

Fourier Analysis Chapter 1 Basic Concepts I. Periodic Function Definition: A function f (x) is called periodic if it is defined for all real x and if there is some positive number p such that f (x + p) ? f (x), for all real x. (1.1) The smallest value of the positive number p is called the period of f (x). cosx and sinx have period of 2?. If f (x) has a period of p, determine the period of f (kx). Solution: Since f (x + p) ? f (x), let x ? kt, then II. Orthogonality Definition: The inner product of the functions f (x) and g(x) with respect to the weighting function r(x) on the interval x([a, b] is f, g r ?. (1.2) If f, g r ? 0, we say that f (x) and g(x) are orthogonal with respect to r(x) on the interval x([a, b]. If r(x) ? 1, we simply write equation (1.2) as f, g ?. (1.3) For simplicity, we confine to the case of the weighting function r(x) ? 1. Definition: The norm of f (x) on the interval x([a, b] is ||f (x)|| ?. If a set of functions ?1(x), ?2(x),…,??n(x), on the interval x([a, b] satisfies ?i (x), ?j (x) ? ?ij || ?i (x) ||2, (1.4) it is called a set of orthogonal functions on the interval x([a, b]. Let ui (x) ?, then ui (x), uj (x) ? ?ij. The set of functions u1(x), u2(x),…, un(x), forms a set of orthonormal functions on the interval [a, b]. Show that 1, cosx, sinx, cos2x, sin2x,…, cosnx, sinnx are orthogonal on the interval x([0, 2?]. Form a set of orthogonal functions from {1, x, x2,…} on the interval (0,1). Solution: Let ?1(x) ? 1, then , III. Even and Odd Functions Definition: A function f (x) is said to be even if, for all x, f (x) ? f (?x), (1.5) and is odd if, for all x f (x) ? ?f (?x). (1.6) In other words, an even function is one whose graph is symmetric about the y-axis and an odd function has graph which is symmetric about the origin. Any function can be written as the sum of an even and odd functions. For f (x) ?[f (x) + f (?x)] +[f (x) ? f (?x)] ? f e (x) + f o (x), where f e (x) ?[f (x) + f (?x)] is even, and f o (x) ?[f (x