By Walter E Thirring

ISBN-10: 0387814965

ISBN-13: 9780387814964

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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Additional resources for A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems

Example text

1 A sequence an of complex numbers is positive-definite if for any sequence of complex numbers ~n E C with only finitely many nonzero terms, L an-m~n~m 2: o. 2 Suppose /-L E M(T) is a positive measure. Then an = /-L(e n ) is a positive-definite sequence. PROOF: Let ~n be a sequence of complex numbers, with finitely many non-zero terms. Set ifJ(O) = L~nen(O). Then, lifJ(OW is a positive function in C(T), and o < /-L(lifJI 2 ) ~ ~ (~Mmen-m(O)) L /-L( en-m)~n~m. 5. 2. 1 (Herglotz) Suppose an is a positive-definite sequence.

The inequality exp is a consequence.

1), under this assumption, B(Vol) = Vol. Thus, if f E V and 'l/J E Vol, we must have 34 CHAPTER 2. HARDY SPACES for all integers n. So all of the Fourier coefficients of the Ll function fi[; must be zero. In other words, if a function 1/J is orthogonal to V, then the set ¥- O} n {O : f(O) ¥- O} has measure zero for every function f E V. e. 0 E X. Any set of the form {O : ¢(0) ¥- O} where ¢ is a nonzero member of V, is an example of such a X. £ > O. Now, let X k be a sequence of such measurable sets such that and set E = UkXk.

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A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems by Walter E Thirring

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