An example of a complete orthonormal system in a Hilbert space of generalized functions

1997 ◽  
Vol 49 (5) ◽  
pp. 812-815
Author(s):  
O. V. Gladkivs’ka
Keyword(s):  
Hilbert Space ◽  
Orthonormal System ◽  
Generalized Functions ◽  
Complete Orthonormal System ◽  
Space Of Generalized Functions

scholarly journals A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model

2013 ◽  
Vol 35 (3) ◽  
pp. 762-834 ◽  
Author(s):  
HAYATO CHIBA
Keyword(s):  
Hilbert Space ◽  
Steady State ◽  
Linear Operator ◽  
Continuous Spectrum ◽  
Imaginary Axis ◽  
Center Manifold ◽  
Generalized Functions ◽  
Kuramoto Model ◽  
Space Of Generalized Functions ◽  
The Kuramoto Model

AbstractThe Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as coupled phase oscillators. In this paper, a bifurcation structure of the infinite-dimensional Kuramoto model is investigated. A purpose here is to prove the bifurcation diagram of the model conjectured by Kuramoto in 1984; if the coupling strength $K$ between oscillators, which is a parameter of the system, is smaller than some threshold ${K}_{c} $, the de-synchronous state (trivial steady state) is asymptotically stable, while if $K$ exceeds ${K}_{c} $, a non-trivial stable solution, which corresponds to the synchronization, bifurcates from the de-synchronous state. One of the difficulties in proving the conjecture is that a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, has the continuous spectrum on the imaginary axis. Hence, the standard spectral theory is not applicable to prove a bifurcation as well as the asymptotic stability of the steady state. In this paper, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid the continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator will be estimated with the aid of the spectral theory on a rigged Hilbert space to prove the linear stability of the steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite-dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimension because of the continuous spectrum on the imaginary axis. These results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto.

scholarly journals ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

2020 ◽  
Vol 8 (2) ◽  
pp. 24-39
Author(s):  
V. Gorodetskiy ◽  
R. Kolisnyk ◽  
O. Martynyuk
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Initial Condition ◽  
Partial Derivatives ◽  
Time Problem ◽  
The Cauchy Problem ◽  
Space Of Generalized Functions ◽  
Derivatives Of

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

scholarly journals Stability of Jensen equations in the space of generalized functions

2004 ◽  
Vol 299 (2) ◽  
pp. 578-586 ◽  
Author(s):  
Linsong Li ◽  
Jaeyoung Chung ◽  
Dohan Kim
Keyword(s):  
Generalized Functions ◽  
Space Of Generalized Functions

scholarly journals An orthonormal system on the construction of the generalized cantor set

1993 ◽  
Vol 16 (4) ◽  
pp. 737-748 ◽  
Author(s):  
Raafat Riad Rizkalla
Keyword(s):  
Orthonormal System ◽  
Walsh System ◽  
Cantor Set ◽  
Complete Orthonormal System

This paper presents a new complete orthonormal system of functions defined on the interval[0,1]and whose supports shrink to nothing. This system related to the construction of the Cantor ternary set. We defined the canonicalmap ξand proved the equivalence between this system and the Walsh system. The generalized Cantor set with any dissection ratio is established and the constructed system is defined in the general case.

scholarly journals Eigenstructure of the equilateral triangle, Part II: The Neumann problem

2002 ◽  
Vol 8 (6) ◽  
pp. 517-539 ◽  
Author(s):  
Brian J. McCartin
Keyword(s):  
Boundary Conditions ◽  
Neumann Problem ◽  
Equilateral Triangle ◽  
Orthonormal System ◽  
Neumann Boundary ◽  
Complete Orthonormal System ◽  
Eigenvalues And Eigenfunctions ◽  
Nodal Lines ◽  
Mathematical Techniques ◽  
The Neumann Problem

Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.

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