傅里葉變換的可視化及應用研究-文庫吧在線文庫

　　

【正文】 nts are related to the by the equations， ，If we set and we haveWith increasing n, converges to zero and the sum converges uniformly to g(x). It follows that also coverfes uniformly to g(x) in the interval , and therefore that itself converges uniformly to h（x） in every closed subinterval not including the point x=0. At this excluded point x=0 all partial sums vanish, so that . Thus the value of the series at the point of discontinuity is also equal to the value of h(x), namely to the arithmetic mean of the righthand and lefthand limits and . As the function h(x) jumps by at x=0. the function jumps by at and is otherwise continuous in the basic interval. Now, if f(x) is a piecewise continuous function which jumps by at the points of the interval , thenis a function which is everywhere continuous and which, along with f(x), has a piecewise continuous first derivative. Therefore, F(x) may be expanded in a Fourier series which converges absolutely and uniformly. But the functionmay also be expanded in a Fourier series, the converfence of which is uniform in every interval not containing any of the points of discontinuity. Thus the theorem formulated at athe beginning of this section is proved pletely.2. Multiple Fourier Series. The trigonometric functions can be used to construct orthogonal systems over “cubes” in higher dimensional spaces. For simplicity we consider only the “plane” case (two dimensions). All our arguments will, however, be valid for any number of dimensions. The functions ， ， ， form an orthogonal system in the square , . The expansion formulas may be written most simply in terms of plex exponentials. If F(s,t) can be expanded in a uniformly convergent double Fourier series, the series iswith The pleteness of the system of functions and thus the pleteness relationfollow by our general theorem on the construction of plete systems in several variables from plete systems in one variable. Futhermore, the Fourier series fot F(s,t) converges absolutely and uniformly if exists and is piecewise continuous,(cf. Subsection 1).3. Order of Magnitude of Fourier Coefficients. If a periodic function f(x) is continuous and has continuous derivatives up to the (h1)st order, if its hth derivative id piecewise continuous, and if the Fourier expansion of f(x) is , then the coefficients , for , can be estimated byWhere c is a constant. Thus, the smoother the function, the more rapidly the coefficients of the series tend to zero. The above relation may be obtained immediately if exoression (15’) for the coefficients is integrated by parts h times.4. Change in Length of Basic Interval. If the function f(x) is periodic with period 2l it may be expanded in the seriesWith，Which may also be written in the form，5. Examples. For simple examples of the application of the theory of Fourier series the reader is referred to elementary texts. Here we shall employ the Fourier series expansion to derive the functional equation of the theta function and a general formula by Poisson. The functional equation of the theta function isTo prove this relation, we set； is evidently a periodic function of y with the period l, which possesses all derivatives with respect to y and which may therefore by expanded in the Fourier series，WithSince the orders of summation and integration may be interchanged for all x0, we obtain for the coefficients : ；For, has the same value along a line parallel to the real axis as along the real axis itself. ( If the Cauchy theorem is applied to the function and to the rectangle with vertices at ,and if T is then made to tend to inifity, the integrals over the vertical sides of the rectangle converge to zero。1. 基本定理的證明：由167。 我們將首先對f(x)為連續而間斷點只出現在導數f’(x)的情況給出證明。因此，在間斷點上級數的值也等于h(x)的值，即左右極限和的算術平均值。但不管怎樣，我們的討論對任意維數都是適用的。：如果函數f(x)是以2l為周期的周期函數，它可以被展開為級數其中，也可以被寫為，5..例子：傅里葉級數理論運用的簡單例子讀者可以參看初等書籍。顯然，如果公式中出現的所有積分都存在，而對t在區間上一致收斂，并且這個級數代表的函數可被展開為傅里葉級數，那么泊松公式就是適用的。這個公式適用于任何分段光滑的函數，這是狄立克雷所證明的。公式中積分的次序由所寫的微分符號所確定。積分 有時又叫做拉普拉斯積分。167。有前一式減去最后一式我們就得到公式（16），它在f(x)為連續時適用。此外，我們增加一條假設，假設積分存在。這個方法在最近用來處理一些出現在數論中的解析函數時，被證明是有效的。 進一步，如果存在且分段連續（見第一小節），則F(s,t)的傅里葉級數一致且絕對收斂。但是函數也可以被展開為傅里葉級數，且在不包含間斷點的任意區間內一致收斂。它的傅里葉展開系數為 ， 如果傅里葉展開對分段連續的函數適用，我們有，為了檢驗這個等式，我們首先構造連續并分段光滑的函數，由上面的討論，這個函數的傅里葉級數一致收斂于g(x)。 為方便起見，我們將假設函數f(x)最初只定義在區間上，然后利用函數方程周期延拓到這區間之外。 it is correct if the passage to the limit can be justified. For real functions f(x) this may also be written in the form（17） A strict proof of the validity of this “Fourier integral formula” is more easily obtained by a direct verification of equation (16) or (17) than by a justification of the pasage to the limit. We start with the integral formula，Where a is an arbitrary positive number. This formula, which is valid for every piecewise smooth function, is due to Dirichlet. From this formula it follows that We assert that the integration over t may be extended from to . For, if Aa, we havewhence, since exists by hypothesis, If we now keep v fixed and let A tend to infinity, it follows thatand, passing to the limit , we obtainThe right