scholarly journals Fourier multipliers on spaces of distributions

1986 ◽  
Vol 29 (3) ◽  
pp. 309-327 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Banach Spaces ◽  
Growth Conditions ◽  
Fourier Multipliers ◽  
Bounded Operators ◽  
Vertical Strip ◽  
One Dimensional ◽  
The One ◽  
Image Position ◽  
Complex Valued

In [8], Rooney defines a class of complex-valued functions ζ each of which is analytic in a vertical strip α(ζ)< Res < β(ζ) in the complex s-plane and satisfies certain growth conditions as |Im s| →∞ along fixed lines Re s = c lying within this strip. These conditions mean that the functionsfulfil the requirements of the one-dimensional Mihlin-Hörmander theorem (see [6, p. 417]) and so can be regarded as Fourier multipliers for the Banach spaces . Consequently, each function gives rise to a family of bounded operators W[ζ,σ] σ ∈(α(ζ),β(ζ)), on , 1<p<∞.

Multiresolution analysis for compactly supported interpolating tensor product wavelets

2015 ◽  
Vol 13 (02) ◽  
pp. 1550010 ◽  
Author(s):  
Tommi Höynälänmaa
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Multiresolution Analysis ◽  
Continuous Functions ◽  
Function Spaces ◽  
One Dimensional ◽  
Space Norm ◽  
The One ◽  
Complex Valued ◽  
Uniformly Continuous

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C0(ℝn, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝn vanishing at infinity and the function spaces Cu(ℝn, K) consisting of bounded and uniformly continuous functions on ℝn. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.

scholarly journals On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

2017 ◽  
Vol 14 (05) ◽  
pp. 1750065 ◽  
Author(s):  
Oktay Veliev
Keyword(s):  
Schrödinger Operator ◽  
Spectral Properties ◽  
Schrodinger Operator ◽  
Symmetric Potential ◽  
One Dimensional ◽  
Symmetric Complex ◽  
The One ◽  
Complex Valued

In this paper, we investigate the spectrum and spectrality of the one-dimensional Schrödinger operator with a periodic PT-symmetric complex-valued potential.

A limit theorem for certain disordered random systems

1994 ◽  
Vol 26 (04) ◽  
pp. 1022-1043 ◽  
Author(s):  
Xinhong Ding
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Limit Theorem ◽  
Random Systems ◽  
Joint Probability ◽  
Dynamical Behaviour ◽  
One Dimensional ◽  
Linear Stochastic Differential Equation ◽  
The One ◽  
Image Position

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in : where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝ p -valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation where P(t, dx dy) is the joint probability law of ξ and X(t).

scholarly journals SUCCESSIVE ITERATION AND POSITIVE SOLUTIONS FOR A p-LAPLACIAN MULTIPOINT BOUNDARY VALUE PROBLEM

2008 ◽  
Vol 49 (4) ◽  
pp. 551-560 ◽  
Author(s):  
BO SUN ◽  
XIANGKUI ZHAO ◽  
WEIGAO GE
Keyword(s):  
Positive Solutions ◽  
Monotone Iterative Method ◽  
One Dimensional ◽  
Multipoint Boundary ◽  
Monotone Iterative ◽  
Existence Of Positive Solutions ◽  
The One ◽  
Image Position ◽  
Multipoint Boundary Condition ◽  
Multipoint Boundary Value Problem

AbstractIn this paper, we study the existence of positive solutions for the one-dimensional p-Laplacian differential equation, subject to the multipoint boundary condition by applying a monotone iterative method.

scholarly journals On the number of eigenvalues in the spectral gap of a Dirac system

1986 ◽  
Vol 29 (3) ◽  
pp. 367-378 ◽  
Author(s):  
D. B. Hinton ◽  
A. B. Mingarelli ◽  
T. T. Read ◽  
J. K. Shaw
Keyword(s):  
Differential Operators ◽  
Spectral Gap ◽  
Equivalence Classes ◽  
Value Functions ◽  
Complex Vector ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Integrable Functions ◽  
The One ◽  
Image Position

We consider the one-dimensional operator,on 0<x<∞ with. The coefficientsp,V1andV2are assumed to be real, locally Lebesgue integrable functions;c1andc2are positive numbers. The operatorLacts in the Hilbert spaceHof all equivalence classes of complex vector-value functionssuch that.Lhas domainD(L)consisting of ally∈Hsuch thatyis locally absolutely continuous andLy∈H; thus in the language of differential operatorsLis a maximal operator. Associated withLis the minimal operatorL0defined as the closure ofwhereis the restriction ofLto the functions with compact support in (0,∞).

scholarly journals Γ-limit for the extended Fisher–Kolmogorov equation

2002 ◽  
Vol 132 (1) ◽  
pp. 141-162 ◽  
Author(s):  
D. Hilhorst ◽  
L. A. Peletier ◽  
R. Schätzle
Keyword(s):  
Fourth Order ◽  
Lyapunov Functional ◽  
Transition Energy ◽  
Kolmogorov Equation ◽  
One Dimensional ◽  
Cahn Equation ◽  
Area Functional ◽  
The One ◽  
Image Position

We consider the Lyapunov functional, of the rescaled Extended Fisher-Kolmogorov equation This is a fourth order generalization of the Fisher–Kolmogorov or Allen–Cahn equation. We prove that if ε → 0, then tends to the area functional in the sense of Γ-limits, where the transition energy is given by the one-dimensional kink of the Extended Fisher–Kolmogorov equation.

Differential equations in Banach spaces and the extension of Lyapunov's method

1963 ◽  
Vol 59 (2) ◽  
pp. 373-381 ◽  
Author(s):  
V. Lakshmikantham
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Existence Theorems ◽  
Linear Operators ◽  
Linear Differential Equations ◽  
Bounded Operators ◽  
Lyapunov's Method ◽  
Linear Differential ◽  
Lyapunov’S Method ◽  
Image Position

The concept of Lyapunov's function is an important tool in studying various problems of ordinary differential equations. In the present paper we shall extend the Lyapunov's method to study some problems of differential equations in Banach spaces. Continuing the theory of one parameter semi-groups of linear and bounded operators founded by Hille and Yoshida, Kato(4) presented some uniqueness and existence theorems for the solutions of linear differential equations of the typewhere A(t) is a given function whose values are linear operators in Banach space. Krasnoselskii, Krein and Soboleveskii (5,6) also considered such equations including non-linear differential equations of the typeMlak (9) obtained some results concerning the limitations of solutions of the latter equation.

Dirac Systems with Discrete Spectra

1987 ◽  
Vol 39 (1) ◽  
pp. 100-122 ◽  
Author(s):  
D. B. Hinton ◽  
J. K. Shaw
Keyword(s):  
Separation Of Variables ◽  
Relativistic Quantum ◽  
Three Dimensional ◽  
Relativistic Quantum Mechanics ◽  
Radial Wave ◽  
One Dimensional ◽  
The One ◽  
Image Position ◽  
Discrete Spectra ◽  
Radial Wave Equation

In this paper we consider the one dimensional Dirac system1.1where αk(x) < 0, λ is a complex spectral parameter, and the remaining coefficients are suitably smooth and real valued. We regard (1.1) as regular at x = a but singular at x = b; in Section 4 we extend our result to problems having two singular endpoints.Equation (1.1) arises from the three dimensional Dirac equation with spherically symmetric potential, following a separation of variables. For the choices p(x) = k/x, αk(x) = 1,p2(x) = (z/x) + c, p1(x) = (z/x) – c, and appropriate values of the constants, (1.1) is the radial wave equation in relativistic quantum mechanics for a particle in a field of potential V = z/x [17]. Such an equation was studied by Kalf [11] in the context of limit point-limit circle criteria, which is one of the matters we consider here.

M-structure in Banach spaces

1967 ◽  
Vol 63 (3) ◽  
pp. 613-629 ◽  
Author(s):  
F. Cunningham
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Measure Space ◽  
Complete Boolean Algebra ◽  
Ordinary Sense ◽  
One Dimensional ◽  
Algebra 2 ◽  
Image Position ◽  
Vector Valued

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfyingfor all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.

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