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Hermitian function
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In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign:

(where the indicates the complex conjugate) for all in the domain of . In physics, this property is referred to as PT symmetry.

This definition extends also to functions of two or more variables, e.g., in the case that is a function of two variables it is Hermitian if

for all pairs in the domain of .

From this definition it follows immediately that: is a Hermitian function if and only if

Motivation

[edit]

Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:[citation needed]

  • The function is real-valued if and only if the Fourier transform of is Hermitian.
  • The function is Hermitian if and only if the Fourier transform of is real-valued.

Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the discrete Fourier transform of a signal (which is in general complex) to be stored in the same space as the original real signal.

  • If f is Hermitian, then .

Where the is cross-correlation, and is convolution.

  • If both f and g are Hermitian, then .

See also

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Hermitian function

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A Hermitian function is a complex-valued function $ f: \mathbb{R} \to \mathbb{C} $ defined on the real numbers that satisfies $ f(x) = \overline{f(-x)} $ for all $ x \in \mathbb{R} $, where $ \overline{\cdot} $ denotes the complex conjugate.[1] This symmetry condition ensures that the real part of $ f $, denoted $ \operatorname{Re}(f) $, is an even function—satisfying $ \operatorname{Re}(f)(x) = \operatorname{Re}(f)(-x) $—while the imaginary part, $ \operatorname{Im}(f) $, is an odd function—satisfying $ \operatorname{Im}(f)(x) = -\operatorname{Im}(f)(-x) $.[1][2] Hermitian functions generalize the concept of even functions to the complex domain and play a fundamental role in mathematical analysis, particularly in Fourier theory and signal processing.[1] A key property is that the Fourier transform of a Hermitian function is real-valued and even, which simplifies computations and interpretations in applications such as wave propagation, quantum mechanics, and image processing.[2] Conversely, the Fourier transform of a real-valued function exhibits Hermitian symmetry, linking the two concepts tightly in transform pairs.[3] This symmetry also arises in the study of self-adjoint operators and Hilbert spaces, where Hermitian properties ensure real eigenvalues and orthogonality in inner products.

Definition and Basics

Formal Definition

A Hermitian function is a complex-valued function on the real line satisfying a symmetry condition that generalizes the notion of even functions to the complex domain. Specifically, for a function $ f: \mathbb{R} \to \mathbb{C} $, it is Hermitian if $ f^(x) = f(-x) $ for all $ x \in \mathbb{R} $, where $ f^ $ denotes the complex conjugate of $ f $. This condition requires basic knowledge of complex conjugation, which maps a complex number $ a + bi $ to $ a - bi $, with $ i $ the imaginary unit, and the negation operation on the argument. This definition extends naturally to multivariate cases. For a function $ f: \mathbb{R}^n \to \mathbb{C} $, it is Hermitian if $ f^*(x_1, \dots, x_n) = f(-x_1, \dots, -x_n) $ for all $ \mathbf{x} = (x_1, \dots, x_n) \in \mathbb{R}^n $. If $ f $ is real-valued, the condition simplifies to $ f(x) = f(-x) $, recovering the standard even function property.[4] It is important to distinguish Hermitian functions, which are scalar functions with real arguments exhibiting this conjugation-based parity symmetry, from Hermitian matrices or forms in linear algebra. The latter are structures where a matrix equals its conjugate transpose or a bilinear form satisfies analogous self-adjointness, serving different roles in inner products and operators.[5]

Relation to Symmetry

Hermitian functions extend the notions of even and odd symmetry from real-valued functions to the complex domain. A function f:R→Cf: \mathbb{R} \to \mathbb{C} satisfying f(−x)=f(x)‾f(-x) = \overline{f(x)} for all xx in its domain has a real part Re⁡(f)\operatorname{Re}(f) that is even, meaning Re⁡(f(−x))=Re⁡(f(x))\operatorname{Re}(f(-x)) = \operatorname{Re}(f(x)), and an imaginary part Im⁡(f)\operatorname{Im}(f) that is odd, meaning Im⁡(f(−x))=−Im⁡(f(x))\operatorname{Im}(f(-x)) = -\operatorname{Im}(f(x)). This decomposition highlights how the Hermitian condition enforces balanced symmetry across the real and imaginary components, generalizing the parity properties observed in real analysis.[6] When a Hermitian function ff is real-valued, the condition simplifies further: since f(x)‾=f(x)\overline{f(x)} = f(x), it follows that f(−x)=f(x)f(-x) = f(x), rendering ff an even function in the classical sense. Conversely, the Hermitian property alone implies the even-odd decomposition of the parts, but for ff to be real-valued everywhere, the odd imaginary part must vanish identically, leaving only the even real part. This interplay underscores the role of Hermitian symmetry in constraining complex functions to real behaviors under specific conditions.[6][7] In inner product spaces such as L2(R)L^2(\mathbb{R}) with the standard Hermitian inner product ⟨f,g⟩=∫−∞∞f(x)g(x)‾ dx\langle f, g \rangle = \int_{-\infty}^{\infty} f(x) \overline{g(x)} \, dx, Hermitian functions preserve symmetry by ensuring that ⟨f,g⟩=⟨g,f⟩‾\langle f, g \rangle = \overline{\langle g, f \rangle} aligns with the function's parity, and specifically, the inner product between two Hermitian functions is real-valued. This property facilitates orthogonality relations akin to those for symmetric real functions, where ⟨f,g⟩=0\langle f, g \rangle = 0 implies mutual perpendicularity without imaginary components. The terminology "Hermitian function" draws from Hermitian matrices and quadratic forms, named after Charles Hermite, who in 1855 introduced complex conjugate forms that exhibit analogous symmetry under conjugation. This concept was formalized for functions in early 20th-century functional analysis, emphasizing a self-adjoint-like symmetry that parallels the positive-definiteness and reality of eigenvalues in matrix theory.[8]

Key Properties

Fourier Transform Characteristics

A fundamental property of Hermitian functions in Fourier analysis is the duality theorem, which asserts that the Fourier transform f^\hat{f} of a Hermitian function ff, satisfying f(−x)=f(x)‾f(-x) = \overline{f(x)}, is real-valued, meaning f^(ω)∈R\hat{f}(\omega) \in \mathbb{R} for all ω∈R\omega \in \mathbb{R}.[9] This result highlights the preservation of reality in the frequency domain under the Hermitian symmetry condition in the spatial domain. The converse of this theorem also holds: if the Fourier transform f^\hat{f} is real-valued, then the original function ff must be Hermitian.[10] The explicit form of the Fourier transform for a Hermitian function ff leverages its symmetry to simplify into a real cosine projection:
f^(ω)=2∫0∞Re⁡(f(x))cos⁡(2πωx) dx. \hat{f}(\omega) = 2 \int_0^\infty \operatorname{Re}(f(x)) \cos(2\pi \omega x) \, dx.
This equation emphasizes how the even real part of ff contributes to the real-valued spectrum via the cosine kernel, with odd components canceling out due to the integral's symmetry.[9] In the multivariate setting, for x∈Rn\mathbf{x} \in \mathbb{R}^n and ff Hermitian such that f(−x)=f(x)‾f(-\mathbf{x}) = \overline{f(\mathbf{x})}, the multidimensional Fourier transform inherits the same reality property, yielding a real-valued f^(ω)\hat{f}(\boldsymbol{\omega}) for all ω∈Rn\boldsymbol{\omega} \in \mathbb{R}^n.[11] This duality can be derived from the standard Fourier transform definition,
f^(ω)=∫−∞∞f(x)e−2πiωx dx, \hat{f}(\omega) = \int_{-\infty}^\infty f(x) e^{-2\pi i \omega x} \, dx,
by exploiting the Hermitian condition f(−x)=f(x)‾f(-x) = \overline{f(x)}. Substituting and taking the complex conjugate yields f^(ω)‾=f^(ω)\overline{\hat{f}(\omega)} = \hat{f}(\omega), confirming the imaginary parts cancel and f^\hat{f} is real; the change of variables u=−xu = -x in the conjugate integral aligns the exponentials to match the original form.[9]

Correlation and Convolution Behaviors

In signal processing and mathematical analysis, the cross-correlation of a Hermitian function ff with another function gg exhibits a notable relationship to convolution. Specifically, the cross-correlation is defined as
(f⋆g)(t)=∫−∞∞f∗(t−τ)g(τ) dτ, (f \star g)(t) = \int_{-\infty}^{\infty} f^*(t - \tau) g(\tau) \, d\tau,
where f∗f^* denotes the complex conjugate of ff. For a Hermitian function ff, satisfying f∗(x)=f(−x)f^*(x) = f(-x), and real-valued gg, this becomes
(f⋆g)(t)=(f∗g)(t)‾, (f \star g)(t) = \overline{(f * g)(t)},
where the convolution is given by
(f∗g)(t)=∫−∞∞f(t−τ)g(τ) dτ. (f * g)(t) = \int_{-\infty}^{\infty} f(t - \tau) g(\tau) \, d\tau.
This relation arises because the Hermitian symmetry introduces conjugation in the alignment of the expressions.[9] Furthermore, in general, the cross-correlation satisfies f⋆g=f∗g~f \star g = f * \tilde{g}, where g~(t)=g(−t)‾\tilde{g}(t) = \overline{g(-t)} is the time-reversed complex conjugate of gg. For real-valued gg, this simplifies to f⋆g=f∗g‾f \star g = f * \overline{g}, with g‾(t)=g(−t)\overline{g}(t) = g(-t), applicable over domains where the integrals converge, such as R\mathbb{R} or suitable intervals for compact support. These relations highlight the role of Hermitian symmetry in maintaining operational consistency across time-domain transformations. The implications of these behaviors extend to linear time-invariant (LTI) systems, where a Hermitian impulse response hh results in outputs that are real-valued for real-valued inputs. In such systems, the response y(t)=(h∗x)(t)y(t) = (h * x)(t) to a real input x(t)x(t) has a Fourier transform Y(ω)=H(ω)X(ω)Y(\omega) = H(\omega) X(\omega), where H(ω)H(\omega) is real-valued, ensuring the output inherits Hermitian symmetry in the frequency domain and is thus real in the time domain. This is particularly useful in filter design, ensuring balanced processing without unintended phase shifts. To outline the proof for the relation f⋆g=f∗g‾f \star g = \overline{f * g} when gg is real and ff is Hermitian, start with the cross-correlation integral:
(f⋆g)(t)=∫−∞∞f∗(t−τ)g(τ) dτ=∫−∞∞f(τ−t)g(τ) dτ, (f \star g)(t) = \int_{-\infty}^{\infty} f^*(t - \tau) g(\tau) \, d\tau = \int_{-\infty}^{\infty} f(\tau - t) g(\tau) \, d\tau,
using f∗(u)=f(−u)f^*(u) = f(-u). Now perform the change of variables σ=t−τ\sigma = t - \tau, so dσ=−dτd\sigma = -d\tau and τ=t−σ\tau = t - \sigma:
(f⋆g)(t)=∫∞−∞f((t−σ)−t)g(t−σ)(−dσ)=∫−∞∞f(−σ)g(t−σ) dσ. (f \star g)(t) = \int_{\infty}^{-\infty} f((t - \sigma) - t) g(t - \sigma) (-d\sigma) = \int_{-\infty}^{\infty} f(-\sigma) g(t - \sigma) \, d\sigma.
Since ff is Hermitian, f(−σ)=f(σ)‾f(-\sigma) = \overline{f(\sigma)}, yielding
(f⋆g)(t)=∫−∞∞f(σ)‾g(t−σ) dσ=∫−∞∞f(σ)g(t−σ) dσ‾=(f∗g)(t)‾, (f \star g)(t) = \int_{-\infty}^{\infty} \overline{f(\sigma)} g(t - \sigma) \, d\sigma = \overline{ \int_{-\infty}^{\infty} f(\sigma) g(t - \sigma) \, d\sigma } = \overline{(f * g)(t)},
as gg is real. Similar substitutions establish the general relation to g~\tilde{g}.[9]

Examples and Applications

Mathematical Examples

A simple example of a Hermitian function is $ f(x) = e^{i k x} $ for a real constant $ k $.[12] The complex conjugate is $ f^(x) = e^{-i k x} $, while $ f(-x) = e^{i k (-x)} = e^{-i k x} $. Thus, $ f(-x) = f^(x) $, confirming it satisfies the defining property.[12] Hermitian functions can also be formed by combining even real-valued polynomials with $ i $ times odd real-valued polynomials. Consider $ f(x) = x^2 + i x $. The complex conjugate is $ f^(x) = x^2 - i x $. Evaluating at the negative argument gives $ f(-x) = (-x)^2 + i (-x) = x^2 - i x $. Therefore, $ f(-x) = f^(x) $, verifying the Hermitian property.[12] This construction works generally because the even real part remains unchanged under $ x \to -x $, while the imaginary part, being odd, acquires a minus sign that matches the conjugation. Another illustrative example is the Gaussian-modulated function $ f(x) = e^{-x^2} (a + i b x) $, where $ a $ and $ b $ are real constants. The real part is $ a e^{-x^2} $, which is even since $ e^{-(-x)^2} = e^{-x^2} $. The imaginary part is $ b x e^{-x^2} $, which is odd because $ (-x) e^{-(-x)^2} = - x e^{-x^2} $. The complex conjugate is $ f^(x) = e^{-x^2} (a - i b x) $, and $ f(-x) = e^{-x^2} (a + i b (-x)) = e^{-x^2} (a - i b x) $. Hence, $ f(-x) = f^(x) $, establishing it as Hermitian.[12] To highlight the requirement, consider a non-example: $ f(x) = i x^2 $. The complex conjugate is $ f^*(x) = -i x^2 $, while $ f(-x) = i (-x)^2 = i x^2 $. Since $ i x^2 \neq -i x^2 $ for $ x \neq 0 $, this function fails the Hermitian condition.[12] The issue arises because the imaginary part $ x^2 $ is even, not odd, violating the necessary symmetry.

Uses in Signal Processing and Physics

In signal processing, Hermitian functions, characterized by their conjugate symmetry $ f(t) = f^*(-t) $, possess real-valued Fourier transforms, which inherently simplifies spectral analysis by eliminating imaginary components.[13] This property reduces computational complexity in algorithms such as the fast Fourier transform (FFT), where conjugate symmetry allows for optimized real-valued implementations that halve the number of operations compared to general complex FFTs.[14] For instance, in transform-based compression methods akin to JPEG but applied to one-dimensional signals like audio waveforms, the real-valued spectra enable efficient encoding with fewer coefficients, lowering storage and transmission demands while preserving signal fidelity.[15] In physics, particularly optics, wavefunctions exhibiting Hermitian symmetry describe symmetric electromagnetic fields in cavities bounded by mirrors, such as Fabry-Pérot resonators, where the symmetry yields real propagation constants for lossless modes. These standing wave patterns, formed by interference of counter-propagating waves, maintain stable resonances without decay, facilitating applications in lasers and precision spectroscopy. In quantum mechanics, position-space wavefunctions exhibiting Hermitian symmetry—where the real part is even and the imaginary part is odd under spatial reflection—simplify the computation of expectation values for the parity operator, as the overlap integral reduces to a straightforward real-valued form without phase complications. This aids in analyzing systems with inversion symmetry, like the harmonic oscillator, where parity eigenvalues directly classify states. This symmetry is prominent in PT-symmetric quantum mechanics and optics, where balanced gain and loss lead to real eigenvalues analogous to Hermitian systems.[16] In engineering applications, such as filter design for audio processing, Hermitian transfer functions enforce conjugate symmetry in the frequency domain, corresponding to real-valued impulse responses that exhibit odd phase symmetry. This phase property minimizes group delay distortion in even-phase filter configurations, like symmetric FIR filters, preserving transient response and reducing artifacts in sound reproduction. Historically, following World War II, these autocorrelation properties of Hermitian signals—yielding real, even functions peaked at zero lag—were pivotal in radar signal design for pulse compression and matched filtering, enhancing target resolution and detection range in early systems developed at institutions like MIT's Lincoln Laboratory.[17] Despite these advantages, not all physical systems produce Hermitian functions; for example, in quantum mechanics, asymmetric potentials disrupt parity invariance, resulting in wavefunctions that lack conjugate symmetry and require more complex numerical methods for analysis.[18] The autocorrelation of Hermitian functions, being real and even, further supports their utility in detection tasks without additional symmetry assumptions.
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