Hilbert transform
[8][9] Some of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in Göttingen.
[12] The Hilbert transform was a motivating example for Antoni Zygmund and Alberto Calderón during their study of singular integrals.
When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of u(t) are respectively shifted by +180° and −180°, which are equivalent amounts.
It is by no means obvious that the Hilbert transform is well-defined at all, as the improper integral defining it must converge in a suitable sense.
[18] (In particular, since the Hilbert transform is also a multiplier operator on L2, Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that H is bounded on Lp.)
The (complex) eigenstates of the Hilbert transform admit representations as holomorphic functions in the upper and lower half-planes in the Hardy space H2 by the Paley–Wiener theorem.
It is possible to work somewhat rigorously with this since compactly supported functions (which are distributions a fortiori) are dense in Lp.
In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space
and its conjugate consist of exactly those L2 functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively.
It is possible to define the Hilbert transform on the space of tempered distributions as well by an approach due to Gel'fand and Shilov,[29] but considerably more care is needed because of the singularity in the integral.
To alleviate such difficulties, the Hilbert transform of an L∞ function is therefore defined by the following regularized form of the integral
[30] Furthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.
A deep result of Fefferman's work[31] is that a function is of bounded mean oscillation if and only if it has the form f + H(g) for some
This harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel
This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc.
[36] Although usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see Paley–Wiener theorem), as well as work by Riesz, Hille, and Tamarkin[37] One form of the Riemann–Hilbert problem seeks to identify pairs of functions F+ and F− such that F+ is holomorphic on the upper half-plane and F− is holomorphic on the lower half-plane, such that for x along the real axis,
The left-hand side of this equation may be understood either as the difference of the limits of F± from the appropriate half-planes, or as a hyperfunction distribution.
The circular Hilbert transform is used in giving a characterization of Hardy space and in the study of the conjugate function in Fourier series.
Another more direct connection is provided by the Cayley transform C(x) = (x – i) / (x + i), which carries the real line onto the circle and the upper half plane onto the unit disk.
The name reflects its mathematical tractability, due largely to Euler's formula.
Applying Bedrosian's theorem to the narrowband model, the analytic representation is:[42] A Fourier transform property indicates that this complex heterodyne operation can shift all the negative frequency components of um(t) above 0 Hz.
presents two causality-based challenges to practical implementation in a convolution (in addition to its undefined value at 0): For a discrete function,
in the region −π < ω < π is given by: The inverse DTFT, using the convolution theorem, is:[46][47] where which is an infinite impulse response (IIR).
[49] This type inherently exhibits responses of zero magnitude at frequencies 0 and Nyquist, resulting in a bandpass filter shape.
An inverse DFT is done on the product, and the transient artifacts at the leading and trailing edges of the segment are discarded.
(an arbitrary parameter) are convolved with the periodic function: When the duration of non-zero values of
Deciding what to delete and the corresponding amount of overlap is an application-dependent design issue.
When the input is a segment of a pure cosine, the resulting convolution for two different values of
Edge effects prevent the result from being a pure sine function (green plot).
The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences.
