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Mathematics for Physics

349 Fourier-Stieltjes Coefficients

A sequence of complex numbers that is used to expand a bounded nondecreasing function on a finite interval as a Fourier-Stieltjes series.

Note: Also written \(c_{n}\). Also called the Fourier-Stieltjes coefficients of \(F\).

definition [d] (Fourier-Stieltjes Coefficients) From Kawata: let \(F(x)\in M(-\pi,\pi)\). Form

  • \(\displaystyle c_{n} = (2\pi)^{-1}\int_{-\pi}^{\pi} e^{-inx}\, dF(x),\quad n=0,\pm 1,\pm 2,\ldots\) .

They are called the Fourier-Stieltjes coefficients of \(F(x)\) and one sometimes uses the representation

  • \(\displaystyle F(x)\sim\sum_{n=-\infty}^{\infty} c_{n}\, e^{inx}\) .

The series is called the Fourier-Stieltjes series.

where

  • \(F\) is a bounded nondecreasing function in \(M(-\pi,\pi)\).
  • \(M(-\pi,\pi)\) is the class of such functions on \([-\pi,\pi]\).
  • \(c_{n}\) is the \(n\)th Fourier-Stieltjes coefficient of \(F\).
  • \(n\) is an integer frequency index.
  • \(x\) is the integration variable.
  • \(i\) is the imaginary unit.
  • \(\displaystyle\int_{-\pi}^{\pi}\cdots\, dF(x)\) is the Lebesgue-Stieltjes integral with respect to \(F\).
  • \((2\pi)^{-1}\) is the normalizing constant.

Integral formula. Each coefficient is a normalized Lebesgue-Stieltjes integral of a complex exponential against \(F\). This principle is used to attach one complex value to every integer frequency.

Fourier-Stieltjes series. The formal series built from the coefficients is written

\[ F(x)\sim\sum_{n=-\infty}^{\infty} c_{n}\, e^{inx} \]

where

  • \(c_{n}\) is the \(n\)th Fourier-Stieltjes coefficient.
  • \(e^{inx}\) is the complex exponential of frequency \(n\).

This principle is used to represent \(F\) by its Fourier-Stieltjes series.

No forced decay. Unlike ordinary Fourier coefficients, the sequence \(c_{n}\) need not tend to zero as \(n\to\pm\infty\). This principle is used to distinguish Fourier-Stieltjes coefficients from classical Fourier coefficients of an integrable density.

349.1 Examples

349.1.1 Simple

Take the unit jump at a point \(a\) with \(-\pi<a<\pi\), restricted to \([-\pi,\pi]\). Then every coefficient is a pure phase.

\[ c_{n} = (2\pi)^{-1} e^{-ina},\quad n=0,\pm 1,\pm 2,\ldots \]

where

  • \(a\) is the jump location in \((-\pi,\pi)\).
  • \(U(x-a)\) is the unit distribution function at \(a\).
  • \(c_{n}\) does not tend to zero as \(n\to\pm\infty\).

349.1.2 General

For a general \(F\in M(-\pi,\pi)\), each coefficient is the Stieltjes integral against the corresponding exponential.

\[ c_{n} = \dfrac{1}{2\pi}\int_{-\pi}^{\pi} e^{-inx}\, dF(x) \]

\[ F(x)\sim\sum_{n=-\infty}^{\infty} c_{n}\, e^{inx} \]

where

  • \(F\) is a bounded nondecreasing function on \([-\pi,\pi]\).
  • \(c_{n}\) is the \(n\)th Fourier-Stieltjes coefficient.
  • the integral is a Lebesgue-Stieltjes integral.

349.2 References

  1. Kawata, T. Fourier Analysis in Probability Theory. Academic Press. Chapter III, §3.2, p. 84 — definition of Fourier-Stieltjes coefficients \(c_{n}=(2\pi)^{-1}\int_{-\pi}^{\pi}e^{-inx}\,dF(x)\); Fourier-Stieltjes series; unit-jump example \(c_{n}=(2\pi)^{-1}e^{-ina}\).