By Tatsuo Kawata
ISBN-10: 0124036503
ISBN-13: 9780124036505
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Extra info for Fourier Analysis in Probability Theory
Sample text
Then the series (1) is called the Fourier series of f(x) and an and £n are called the Fourier cosine coefficients and Fourier sine coefficients^ respectively. Together they are called the Fourier coefficients. In this case, the cn are given by cn = (2π)"1 Γ f(x)e-™* dx, J -π n = 0, ± 1 , ± 2 , . . (9) They are called Fourier coefficients in complex form. If f(x) is an even function, then bn = 0, w = 1, 2 , . . , and aw turns out to be and if f(x) happens to be an odd function, then In order to indicate that (1) is the Fourier series of /(#), we use the notation or, in complex form, anybn, and cn being given by (7), (8), and (9), respectively.
R j = (2π)-ι Γ l o g | / ( r ^ ) | Λ , J —71 (1) where zeros are counted as many times as their multiplicities. Equation (1) is also called Jensen's formula. If n(x) is the number of zeros of f(z) in | z | ^ x, then (1) is written in the form f [n(x)lx] dx = {2η)-1 Γ l o g | / ( r ^ ) | άθ — log|/(0) | for r < R. (2) An immediate consequence of (1) or (2) is (2π)-ι f provided/(0)^0. l o g | / ( « « ) | dQ ^ log|/(0) |, (3) 32 I. INTRODUCTION In the Jensen theorem above, the domain is a circle.
1 (Jensen). Let f(z) be analytic in | z \ < R. Suppose that /(0) φ 0 and let rx ^ r2 ^ · · · be the moduli of zeros of f(z) in | z | < R. Then for rn ^ r ^ r n+1> log[r* | / ( 0 ) | / Γ Λ . . r j = (2π)-ι Γ l o g | / ( r ^ ) | Λ , J —71 (1) where zeros are counted as many times as their multiplicities. Equation (1) is also called Jensen's formula. If n(x) is the number of zeros of f(z) in | z | ^ x, then (1) is written in the form f [n(x)lx] dx = {2η)-1 Γ l o g | / ( r ^ ) | άθ — log|/(0) | for r < R.