a square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is...

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...

transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier...

Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses...

Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals...

of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric...

The result holds as stated provided f {\displaystyle f} is a square-integrable function or, more generally, in Lp space L 2 [ − π , π ] . {\displaystyle...

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...

related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz...

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...

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection...

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...

not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent...

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...

series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or...

integrable. The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function g...

{\displaystyle f} is a square integrable function on X , {\displaystyle X,} and g {\displaystyle g} is a square integrable function on Y , {\displaystyle...

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...

one of the steps involved in the geometric quantization of a square-integrable functions on a symplectic manifold. Kenji FUKAYA, Floer homology of Lagrangian...

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...

characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution)...

Square-integrable function: the square of its absolute value is integrable. Relative to measure and topology: Locally integrable function: integrable around every...

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...

Analytic function Quasi-analytic function Non-analytic smooth function Flat function Bump function Differentiable function Integrable function Square-integrable...

given a domain Ω in n-dimensional Euclidean space Rn, then the square-integrable function k : Ω × Ω → C belonging to L2(Ω×Ω) such that ∫ Ω ∫ Ω | k ( x ...

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...

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...

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...

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...

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]...