By Boris Moiseevich Levitan, Vasiliĭ Vasilʹevich Zhikov

ISBN-10: 0521244072

ISBN-13: 9780521244077

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**Example text**

As a Theorem of the argument 49 result, co(t) is defined as a single-valued function with accuracy up to an integer multiple of 27r. It will be defined completely if as well as continuity we demand that —7r - -. co(0) <7r. The function co(t) is called the argument of f(t) and is denoted by arg f(t). The next theorem has many important applications. Theorem of the argument for an almost periodic function. Suppose that a numerical complex-valued almost periodic function satisfies condition (25). Then arg f(t)= ct +OM, ow (26) where c is a constant and is a continuous almost periodic function.

Ki2A 2 + • • • + k' flit,, = k" iA i + k"2/1 2 + • • • + k"„A n, then by the condition in the theorem, V ie ' + k 12 02 + • • • + le n en r--Ek" 1 61 1 + k"26)2 + • • • +k"nen (mod 27r). Therefore, the coefficients of similar terms have the same arguments. ,k n = {F(1, 1, .. , 1) } P = (n + 1)P. v (5) Now we assume that stu If(t)1= k

Fi,... , AN = TI (N) ' 4, In this notation AI N Q (0 =n=1 E an exp (ih (n ) ' At). We consider the function of m real variables xi, X2,. . , xrn N G(g)= E an exp (ih (n ) • g), n=1 where ±- = (x i , . . , x,n ). By Theorem 5 of this chapter we have k. We denote by arg G(g) an arbitrary continuous branch of the argument of G(g). For every l =1, . . , m there exists an integer gi, independent of i by the continuity of arg G(g), such that arg G(.. , xi +27, . . ) — arg G(. . , Xi,. ) = gi . 27r.

### Almost Periodic Functions and Differential Equations by Boris Moiseevich Levitan, Vasiliĭ Vasilʹevich Zhikov

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