Titchmarsh theorem

Not to be confused with the Titchmarsh convolution theorem.

In the mathematical field of Fourier analysis, the Titchmarsh theorem, named after Edward Charles Titchmarsh, gives necessary and sufficient conditions for a complex-valued square-integrable function F(x) on the real line to be the boundary value of a function in the Hardy space H²(U) of holomorphic functions in the upper half-plane U. The theorem is named for Edward Charles Titchmarsh, who proved the theorem in Titchmarsh (1948, Theorem 95).

The theorem states that the following conditions for a complex-valued square-integrable function F : R → C are equivalent:

• F(x) is the limit as z → x of a holomorphic function F(z) in the upper half-plane such that

$\int _{-\infty }^{\infty }|F(x+iy)|^{2}\,dx<K.$

• −Im(F) is the Hilbert transform of Re(F), where Re(F) and Im(F) are real-valued functions with F = Re(F) + i Im(F).

• The Fourier transform ${\mathcal {F}}(F)(x)$ vanishes for x < 0.

A weaker result is true for functions of class L^p for p > 1 (Titchmarsh 1948, Theorem 103). Specifically, if F(z) is a holomorphic function such that

$\int _{-\infty }^{\infty }|F(x+iy)|^{p}\,dx<K$

for all y, then there is a complex-valued function F(x) in L^p(R) such that F(x + iy) → F(x) in the L^p norm as y → 0 (as well as holding pointwise almost everywhere). Furthermore,

$F(x)=f(x)-ig(x)\,$

where ƒ is a real-valued function in L^p(R) and g is the Hilbert transform (of class L^p) of ƒ.

This is not true in the case p = 1. In fact, the Hilbert transform of an L¹ function ƒ need not converge in the mean to another L¹ function. Nevertheless (Titchmarsh 1948, Theorem 105), the Hilbert transform of ƒ does converge almost everywhere to a finite function g such that

$\int _{-\infty }^{\infty }{\frac {|g(x)|^{p}}{1+x^{2}}}\,dx<\infty .$

This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc.

See also

• Paley–Wiener theorem

References

• Duren, P. (1970), Theory of $H^{p}$-Spaces, New York: Academic Press.
• Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.) (published 1986), ISBN 978-0828403245 {{citation}}: ISBN / Date incompatibility (help).
• Weisstein, Eric W. "Titchmarsh theorem". MathWorld.