Topic
Square-integrable function
About: Square-integrable function is a research topic. Over the lifetime, 1477 publications have been published within this topic receiving 23633 citations.
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Papers
TL;DR: In this article, the authors studied square integrable coefficients of an irreducible representation of the non-unimodular $ax + b$-group and obtained explicit expressions in the case of a particular analyzing family that plays a role analogous to coherent states (Gabor wavelets) in the usual $L_2 $ -theory.
Abstract: An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $\Gamma $-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.
3,804 citations
TL;DR: In this article, the authors consider a mean-field stochastic differential equation, also called the McKean-Vlasov equation, whose coefficients depend on both the solution Xt,xsXst,x and its law.
Abstract: In this paper, we consider a mean-field stochastic differential equation, also called the McKean–Vlasov equation, with initial data (t,x)∈[0,T]×Rd(t,x)∈[0,T]×Rd, whose coefficients depend on both the solution Xt,xsXst,x and its law. By considering square integrable random variables ξξ as initial condition for this equation, we can easily show the flow property of the solution Xt,ξsXst,ξ of this new equation. Associating it with a process Xt,x,PξsXst,x,Pξ which coincides with Xt,ξsXst,ξ, when one substitutes ξξ for xx, but which has the advantage to depend on ξξ only through its law PξPξ, we characterize the function V(t,x,Pξ)=E[Φ(Xt,x,PξT,PXt,ξT)]V(t,x,Pξ)=E[Φ(XTt,x,Pξ,PXTt,ξ)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of mean-field type, involving the first- and the second-order derivatives of VV with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first- and second-order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Ito formula. In our approach, we use the notion of derivative with respect to a probability measure with finite second moment, introduced by Lions in [Cours au College de France: Theorie des jeu a champs moyens (2013)], and we extend it in a direct way to the second-order derivatives.
327 citations
TL;DR: In this article, it was shown that the solution of the Stokes problem has square integrable second derivatives provided that the domain is a convex polygon in the plane and the nonhomogeneous term is square integral.
327 citations
TL;DR: In this paper, the requirements concerning the scores-generating function are relaxed to a minimum: they assume that this function is a difference of two non-decreasing and square integrable functions.
Abstract: This is a straightforward continuation of Hajek (1968). We provide a further extension of the Chernoff-Savage (1958) limit theorem. The requirements concerning the scores-generating function are relaxed to a minimum: we assume that this function is a difference of two non-decreasing and square integrable functions. Thus, in contradistinction to Hajek (1968), we dropped the assumption of absolute continuity. The main results are accumulated in Section 2 without proofs. The proofs are given in Sections 4 through 7. Section 3 contains auxiliary results.
297 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 7 |
| 2024 | 13 |
| 2023 | 48 |
| 2022 | 106 |
| 2021 | 48 |
| 2020 | 62 |