Young's theorem (Fourier coefficients)

The set of Young on Fourier coefficients is a classic theorem of mathematical sub-region of the harmonic analysis . It goes back to the English mathematician William Henry Young and deals with the question of which zero sequences occur as sequences of Fourier coefficients of Lebesgue integrable real functions . As the mathematician Jürgen Elstrodt notes in his textbook Measurement and Integration Theory , this question is considered a difficult problem in the theory of Fourier series . The sentence mentioned is one of the most beautiful sentences by Young .

Formulation of the sentence

The sentence can be formulated as follows:

Is a convex zero sequence of positive real numbers , ${\displaystyle \left(a_{n}\right)_{n=1,2,3,\ldots }}$

so the series formed from this is the Fourier series of a Lebesgue integrable even function , ${\displaystyle \sum _{n=1}^{\infty }{\bigl (}a_{n}\cos {nt}{\bigr )}}$

that is, there is a Lebesgue integrable even function such ${\displaystyle f\colon [-\pi ,\pi ]\to \mathbb {R} }$

that the equation is always true . ${\displaystyle n=1,2,3,\ldots }$ ${\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }{\bigl (}f(t)\cos {nt}{\bigr )}\mathrm {d} t}$

Explanations

• In sequence theory, a delta operator is used , which works in such a way that it assigns the sequence of successive differences to a sequence of real numbers (or a sequence of complex numbers or, more generally, a sequence in an Abelian group ) . This goes over to the new episode . ${\displaystyle \left(z_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \left(z_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \left(\Delta {z_{n}}\right)_{n=1,2,3,\ldots }=\left(z_{n}-z_{n+1}\right)_{n=1,2,3,\ldots }}$
• The double application of the delta operator to the sequence results in the further sequence . ${\displaystyle \left(z_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \left({\Delta }^{2}{z_{n}}\right)_{n=1,2,3,\ldots }=\left(z_{n}-2z_{n+1}+z_{n+2}\right)_{n=1,2,3,\ldots }}$
• A sequence of real numbers is called a convex sequence if the inequality is always satisfied. ${\displaystyle n=1,2,3,\ldots }$ ${\displaystyle {\Delta }^{2}{z_{n}}\geq 0}$
• The mentioned convexity condition means that the inequality always exists. ${\displaystyle n=1,2,3,\ldots }$ ${\displaystyle z_{n+1}\leq {\frac {z_{n}+z_{n+2}}{2}}}$

literature

• Jürgen Elstrodt : Measure and integration theory (=  Springer textbook - basic knowledge of mathematics ). 7th, corrected and updated edition. Springer-Verlag , Heidelberg (inter alia) 2011, ISBN 978-3-642-17904-4 .
• Živorad Tomovski : Convergence and integrability for some classes of trigonometric series . In: Dissertationes Mathematicae (Rozprawy Matematyczne) . tape 420 , 2003 ( MR2030824 ).
• Antoni Zygmund : Trigonometric Series. Volumes I and II . Reprinting of the 1968 Version of the Second Edition. 2nd Edition. Cambridge University Press , Cambridge, London, New York, Melbourne 1977, ISBN 0-521-07477-0 ( MR0617944 ).

Individual evidence

1. ↑ ^{a b} Jürgen Elstrodt: Measure and integration theory. 2011, p. 138
2. ↑ Živorad Tomovski: Convergence and integrability for some classes of trigonometric series. , Dissertationes Mathematicae 420, p. 1 ff, p. 6
3. ^ Antoni Zygmund: Trigonometric Series. Vol. I. 1977, p. 93 ff