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Function whose squared absolute value has finite integral
a square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is
Square-integrable_function
Mathematical description of quantum state
properties of the function spaces of wave functions. In this case, the wave functions are square integrable. One can initially take the function space as the
Wave_function
of all square integrable holomorphic functions on a domain D in Cn. In detail, let L2(D) be the Hilbert space of square integrable functions on D, and
Bergman_kernel
Mathematical transform that expresses a function of time as a function of frequency
(complex-valued) functions in L1(ℝ), it is not well-defined for other integrability classes, most importantly the space of square-integrable functions L2(ℝ)
Fourier_transform
Operation in calculus
is equivalent to the Riemann integral. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of
Integral
Type of vector space in math
of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric
Hilbert_space
Decompositions of inner product spaces into orthonormal bases
Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses
Generalized_Fourier_series
Generalized function whose value is zero everywhere except at zero
almost everywhere, then f {\displaystyle f} is integrable if and only if g {\displaystyle g} is integrable and the integrals of f {\displaystyle f} and
Dirac_delta_function
Basic result in harmonic analysis on compact topological groups
C(G) of continuous complex-valued functions on G, and thus also in the space L2(G) of square-integrable functions. The second part asserts the complete
Peter–Weyl_theorem
Result in Fourier analysis
f̂(n)=12π∫−ππf(x)e−inxdx. The result holds as stated, provided f is a square-integrable function or, more generally, in Lp space L2[−π,π]. A similar result is
Parseval's_identity
Special mathematical functions defined on the surface of a sphere
also dense in the space L2(Sn−1) of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into
Spherical_harmonics
Square-integrable function: the square of its absolute value is integrable. Relative to measure and topology: Locally integrable function: integrable around every
List_of_types_of_functions
Mathematical theorem
not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent
Paley–Wiener_theorem
Function for integral Fourier-like transform
series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or
Wavelet
Topics referred to by the same term
Square-summable may refer to: Square-integrable functions Square-summable sequences; see Hilbert space § Sequence spaces This disambiguation page lists
Square-summable
Tensor product space endowed with a special inner product
{\displaystyle f} is a square integrable function on X , {\displaystyle X,} and g {\displaystyle g} is a square integrable function on Y , {\displaystyle
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
is not itself in general representable by a square-integrable function. Given a square-integrable function ψ ∈ L 2 ( R ) {\displaystyle \psi \in L^{2}(\mathbb
Dual_wavelet
First known wavelet basis
Alfréd Haar. Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval [0, 1]
Haar_wavelet
Mathematical theorem
related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz
Riesz–Fischer_theorem
Specific linear basis (mathematics)
orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval [ − 1 , 1 ] {\displaystyle [-1,1]} can be expressed
Orthonormal_basis
Special function of two variables
real-analytic analogue of a classical elliptic modular function. Note that E(z,s) is not a square-integrable function of z with respect to the invariant Riemannian
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
Product of a number by itself
that the square function is an even function. The squaring operation defines a real function called the square function or the squaring function. Its domain
Square_(algebra)
Mathematical theorem about functions
integrable. The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function g
Fourier_inversion_theorem
Type of convergence in Hilbert spaces
\int _{0}^{2\pi }\sin(nx)\cdot g(x)\,dx.} tends to zero for any square-integrable function g {\displaystyle g} on [ 0 , 2 π ] {\displaystyle [0,2\pi ]} when
Weak convergence (Hilbert space)
Weak_convergence_(Hilbert_space)
Theorem in mathematics
are two complex-valued functions on R {\displaystyle \mathbb {R} } of period 2 π {\displaystyle 2\pi } that are square integrable (with respect to the Lebesgue
Parseval's_theorem
Relative importance of certain frequencies in a composite signal
total energy is finite (i.e. x ( t ) {\displaystyle x(t)} is a square-integrable function) allows applying Parseval's theorem (or Plancherel's theorem)
Spectral_density
Function spaces generalizing finite-dimensional p norm spaces
g⟩=∫Sf(x)g(x)‾dμ(x). Functions in L2 are sometimes called square-integrable functions, quadratically integrable functions or square-summable functions, but sometimes
Lp_space
Model of quantum computing
complex-valued functions on {0,1}n and is naturally an inner product space. ℓ 2 {\displaystyle \ell ^{2}} means the function is a square-integrable function. This
Quantum_circuit
Bounded linear operator
complex-valued square-integrable functions on the interval [0,1]. On the subspace C[0,1] of continuous functions it represents indefinite integration. It is the
Volterra_operator
Mathematical theorem
{\displaystyle \varphi } can range through the space of real-valued square-integrable functions L2[a, b]; however, in many cases the associated reproducing kernel
Mercer's_theorem
Property of certain dynamical systems
characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution)
Integrable_system
Construction for adding objects to a Hilbert space
description of a non-relativistic particle using the Hilbert space of square-integrable functions on the real line, eigenstates of the position and momentum operators
Rigged_Hilbert_space
Proof that every structure with certain properties is isomorphic to another structure
representation theorem states that a Hilbert space, such as the square-integrable function space L2(X) on a manifold X, any linear functional F is equal
Representation_theorem
Mathematics of real numbers and real functions
distance between two functions is essentially the average distance between their values. Spaces of square integrable measurable functions form a metric space
Real_analysis
Integral transform and linear operator
theorem states that the following conditions for a complex-valued square-integrable function F : R → C {\displaystyle F:\mathbb {R} \to \mathbb {C} } are equivalent:
Hilbert_transform
Mathematical function
Without reference to any of these particularities, let f be a square-integrable function of physical space, and let ℋ represent Fourier transformation
Slepian_function
One of Fredholm's theorems in mathematics
set of all square-integrable functions on Ω {\displaystyle \Omega } whose weak first and second derivatives exist and are square-integrable, and which
Fredholm_alternative
Multi particle state space
square-integrable functions on a space X {\displaystyle X} with measure μ {\displaystyle \mu } (strictly speaking, the equivalence classes of square integrable
Fock_space
Approximation method in statistics
dimension to prove a least-squares estimator can be interpreted as a measure on the space of square-integrable functions. In some contexts, a regularized
Least_squares
Map from a vector space to its underlying field
space L 2 ( [ − π , π ] ) {\displaystyle L^{2}([-\pi ,\pi ])} of square integrable functions on [ − π , π ] : {\displaystyle [-\pi ,\pi ]:} f ↦ ⟨ f , g ⟩
Functional_(mathematics)
Methods of calculating definite integrals
of accuracy. If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods
Numerical_integration
Foliation of symplectic manifolds
one of the steps involved in the geometric quantization of a square-integrable functions on a symplectic manifold. Kenji FUKAYA, Floer homology of Lagrangian
Lagrangian_foliation
Element of a basis for a function space
August 2026. Sines and cosines form an orthonormal basis for square-integrable functions on a bounded interval. For the interval [0,1], one such basis
Basis_function
Millennium Prize Problem
2(Ω)⊂L2(Ω) is the Sobolev space of square-integrable functions, for which up to k partial differentiations are also square-integrable. Through the restriction of
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
window function (or Gabor atom) g either in time or frequency for an exact Gabor frame (Riesz Basis). Suppose g is a square-integrable function on the
Balian–Low_theorem
Complex-differentiable (mathematical) function
each variable separately. More generally, a function of several complex variables that is square integrable over every compact subset of its domain is
Holomorphic_function
Mathematical technique used in data compression and analysis
mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet
Wavelet_transform
Mathematical theorem
space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G. The simplest
Selberg_trace_formula
French mathematician (1878–1973)
discovered the representation theorem in the space of Lebesgue square integrable functions. He is often referred to as the founder of the theory of abstract
René_Maurice_Fréchet
Group of rotations in 3 dimensions
square integrable complex-valued functions on the sphere. The inner product on this space is given by If f is an arbitrary square integrable function
3D_rotation_group
Algebraic operation on coordinate vectors
{\displaystyle i} by the function/vector u {\displaystyle u} . This notion can be generalized to square-integrable functions: just as the inner product
Dot_product
Mathematical inequality relating the derivative of a function to its covariant derivative
usual Hilbert space of square-integrable functions, and H1(ℝn) the Sobolev space of square-integrable functions with square-integrable derivatives. Let f
Diamagnetic_inequality
Noncommutative geometric structure
infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional
Singular_trace
Decomposition of periodic functions
square integrable, then the Fourier series of s {\displaystyle s} converges absolutely and uniformly to s ( x ) {\displaystyle s(x)} . If a function is
Fourier_series
Description of a quantum-mechanical system
not square-integrable. Likewise a position eigenstate would be a Dirac delta distribution, not square-integrable and technically not a function at all
Schrödinger_equation
Recipe for constructing a quantum analog of a classical physical theory
Hilbert space" of square-integrable functions on M {\displaystyle M} (with respect to the Liouville volume measure). For each smooth function f {\displaystyle
Geometric_quantization
Sum of inverse squares of natural numbers
}^{2}(0,1)} of L2 periodic functions over ( 0 , 1 ) {\displaystyle (0,1)} (i.e., the subspace of square-integrable functions which are also periodic),
Basel_problem
Calculus of stochastic differential equations
is finite for all t. For any such square integrable martingale M, the quadratic variation process [M] is integrable, and the Itô isometry states that
Itô_calculus
Russian mathematician
problem in the list, on the convergence of the Fourier series for a square-integrable function, came to be called Luzin's conjecture and was solved by Lennart
Nikolai_Luzin
Formula for the derivative of an inverse function
problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function. y=ex (for real x) has inverse
Inverse_function_rule
Systematic procedure of turning a classical theory into a quantum one
once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase
Quantization_(physics)
Mathematical manifold theory
=\|\omega \|^{2}<\infty ,} then the integrand is a real valued, square integrable function on M, evaluated at a given point via its point-wise norms, ‖ ω
Hodge_theory
*-algebra of bounded operators on a Hilbert space
Hilbert space L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} of square-integrable functions. The algebra B ( H ) {\displaystyle {\mathcal {B}}({\mathcal
Von_Neumann_algebra
Functor type
v\in H} . For example, the continuous linear functionals on the square-integrable function space H = L 2 ( R ) {\displaystyle H=L^{2}(\mathbb {R} )} are
Representable_functor
Matrix of inner products of vectors
in ℂn, G=V†V, where V† is the conjugate transpose of V. Given square-integrable functions {ℓi(⋅),i=1,…,n} on the interval [t0,tf], the Gram matrix G=[Gij]
Gram_matrix
Number, approximately 3.14
satisfies Wirtinger's inequality: for a function f:[0,1]→ℂ with f(0) = f(1) = 0 and f, f′ both square integrable, we have: π2∫01|f(x)|2dx≤∫01|f′(x)|2dx
Pi
Length in a vector space
) , {\displaystyle (X,\Sigma ,\mu ),} which consists of all square-integrable functions, this inner product is ⟨ f , g ⟩ L 2 = ∫ X f ( x ) ¯ g ( x )
Norm_(mathematics)
Theorem on boundedness of symmetric operators
other words, it will map some functions in L2(R) to functions that are no longer square integrable. One such function could be ψ ( x ) = 1 π 1 + x 2
Hellinger–Toeplitz_theorem
Analytic function Quasi-analytic function Non-analytic smooth function Flat function Bump function Differentiable function Integrable function Square-integrable
List_of_real_analysis_topics
Representations of finite groups, particularly on vector spaces
} This may be done, by proving that there exists no non-zero square integrable function on G {\displaystyle G} orthogonal to all the irreducible characters
Representation theory of finite groups
Representation_theory_of_finite_groups
Concepts from linear algebra
the Schrödinger equation, one looks for ψE within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar
Eigenvalues_and_eigenvectors
Two-volume set of books by Antoni Zygmund
Carleson's theorem about almost everywhere pointwise convergence for square-integrable functions.[citation needed] Zygmund, Antoni (1935). Trigonometrical series
Trigonometric_Series
Concept in topology
classes of square integrable functions that differ on sets of measure zero, rather than simply the vector space of square integrable functions that the
Kolmogorov_space
Function used in signal processing
argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero. Window functions are used in spectral
Window_function
Expected value of a quantum measurement
the space of square-integrable functions on the real line. Vectors ψ ∈ H {\displaystyle \psi \in {\mathcal {H}}} are represented by functions ψ ( x ) {\displaystyle
Expectation value (quantum mechanics)
Expectation_value_(quantum_mechanics)
Objects that generalize functions
do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Branch of mathematics
which are Banach spaces, and especially the L2 space of square-integrable functions, which is the only Hilbert space among them. Functional analysis
Linear_algebra
Concept in mathematics
{\displaystyle L^{2}[0,1]} of the square-integrable functions on the unit interval. Both are systems of bounded functions, unlike, say, the Haar system or
Walsh_function
Certain vector fields are the sum of an irrotational and a solenoidal vector field
Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and A ∈ H(curl, Ω)
Helmholtz_decomposition
Variant Fourier transforms
factors (see Fourier transform § Unitarity and definition for square integrable functions for discussion), other authors also define the cosine transform
Sine_and_cosine_transforms
Method of evaluating certain integrals along paths in the complex plane
integration is used to study complex-valued functions that are holomorphic in a region. Contour integration is closely related to the calculus of residues
Contour_integration
Generalization of the concept of directional derivative
dt} Let X {\displaystyle X} be the Hilbert space of square-integrable functions on a Lebesgue measurable set Ω {\displaystyle \Omega } in the
Gateaux_derivative
Conditions for switching order of integration in calculus
Use the condition that the functions are integrable to write them as the difference of two positive integrable functions and apply Tonelli's theorem
Fubini's_theorem
In functional analysis, a Hilbert space
L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} is the set of square integrable functions, and F ( ω ) = ∫ − ∞ ∞ f ( t ) e − i ω t d t {\textstyle F(\omega
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Linear map or polynomial function of degree one
Discontinuous linear map Linear least squares "The term linear function means a linear form in some textbooks and an affine function in others." Vaserstein 2006
Linear_function
Lie group of complex numbers of unit modulus; topologically a circle
for L2(𝕋) with respect to normalized Haar measure. Thus any square-integrable function on 𝕋 has an expansion f=∑n=−∞∞f^(n)en with convergence in L2
Circle_group
Modes of vibration in mathematics
The eigenspaces are orthogonal in the space of square-integrable functions, and consist of smooth functions. In fact, the Dirichlet Laplacian has a continuous
Dirichlet_eigenvalue
System with an infinite-dimensional state-space
is L2(0, ∞;U), the space of (equivalence classes of) U-valued square integrable functions on the interval (0, ∞), but other choices such as L1(0, ∞;U)
Distributed_parameter_system
Topics referred to by the same term
Volterra operator, a bounded linear operator on the space of square integrable functions, the operator corresponding to an indefinite integral Volterra
Volterra_(disambiguation)
Calculus of functions generalization
a function. Note each locally integrable function u defines the linear functional φ↦∫uφdx on Cc∞(M) and, moreover, each locally integrable function can
Calculus_on_Euclidean_space
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Functions on special groups related to their matrix representations
the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G. Matrix coefficients of representations of Lie groups turned
Matrix_coefficient
Chaotic map from the unit square into itself
transfer operator is unitary on the Hilbert space of square-integrable functions on the unit square. The spectrum is continuous, and because the operator
Baker's_map
Objects extending the notion of functions
trigonometric series, which were not necessarily the Fourier series of an integrable function. These were disconnected aspects of mathematical analysis at the
Generalized_function
Topics referred to by the same term
name of the O2 Academy Liverpool The L2 space of square-integrable functions L2 norm The ℓ2 space of square-summable sequences L2 cohomology, a cohomology
L2
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
is square integrable on the complement of a neighbourhood of P. Moreover, if h is any real-valued smooth function on X with dh square integrable and
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Mathematical operator in real and harmonic analysis
harmonic analysis. The operator takes a locally integrable function f:ℝd→ℂ and returns another function Mf:ℝd→[0,∞], where Mf(x) is the supremum of the
Hardy–Littlewood maximal function
Hardy–Littlewood_maximal_function
Motivating example in mathematical study
H^{1}(D)} of square integrable functions with domain D {\displaystyle D} whose weak first derivatives is square integrable, containing those functions with the
Obstacle_problem
Operator on a Hilbert space that shifts basis vectors
smooth functions on the unit interval, but has a continuous spectrum (on the unit disk), when acting on the Hilbert space of square-integrable functions. When
Unilateral_shift_operator
Integration method to calculate volume
to shell integration, which integrates along an axis perpendicular to the axis of revolution. If the function to be revolved is a function of x, the
Disc_integration
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SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
SQUARE INTEGRABLE-FUNCTION
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