By Raphael Salem
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Additional info for Algebraic numbers and Fourier analysis (Heath mathematical monographs)
Tn)zn} J 10 (R 29 1 + 'f "11' > ~. T- n + k-l I:1 'Y1'(af + a/ n) and, since I a1'n + a/ n I < 2, l,-l I I: 'YiCa/ + a/ ) I < n 1 k-l 2 I: "11' < i· 1 Take now for m the smallest integer such that }J. I.. , m = 1og T + 1. T n}. T n} zn} m > ~. Tn)Zn I < m- +}J. T . 4 T2 - 1 Thus where A is a function of }J. and theorem. T only, which proves the second part of our 30 Applications to Power Series,. Another Class of Algebraic Integers 4. Properties of the numbers of the class T THEOREM IV. Every number of the class S t is a limit point of numbers of the class T on both sides .
But it is enough that it should belong to H2 in order that () should belong to S. Applications to Power Series,. Another Class of Algebraic Integers 27 The corresponding theorem for the class T is the following one. Ill. Let T be a real number> 1. A necessary and sufficient condition for the existence of a real jJ. ~ 0 such that the power series t THEOREM should have its real part bounded above (without belonging to the class H2) for I z / < 1 is that T should belong to the class T. Then jJ. is algebraic and belongs to the field of T.
28 Applications to Power Series,. Another Class of Algebraic Integers The determinant 1 u Ll = • •• U k- 1 k-l 1 PI . PI 1 ••• Pk-l Pho-l k-l being not zero, we can, by Minkowski's theorem (as given at the beginning of Chapter VI and Appendix, 9), find rational integers AI, ... , Ak, such that the number o= A1uk - 1 + ... ••, (3k-1 all less than 1 in absolute value. In other words, o is a number of the class S belonging to the field of u. Its conjugates are all real. Take now J,L = 02h and l'j = (3lh, h being a positive integer such that 1'1 + + ...
Algebraic numbers and Fourier analysis (Heath mathematical monographs) by Raphael Salem