Integration theory for Hammerstein operators

We develop in this article a strong nonlinear integral and obtain a Riesz-type theorem (utilizing this integral) for the class of (nonlinear) Hammerstein operators. The integral is extended to the class $\text{M}$_E($\text{B}$) of E-valued totally $\text{B}$-measurable functions and convergence theorems are studied. Then an exchange of information is carried out between the operators and the corresponding set functions; for example, the implication of the operator being compact or unconditionally summing is drawn. In the latter case it is shown that the representing set function is analogous to strongly bounded set functions. A vast body of literature exists for both of these concepts.     

Convergence and stability of a collocation method for the generalized airfoil equation

We establish the mean-square convergence and numerical stability of a collocation method for solving a class of Cauchy-singular integral equations. The method uses the Jacobi polynomials {P^{$\text{1}\text{2}$, $\text{-1}\text{2}$}_n} as basis elements and the zeros of the polynomial P^{$\text{-1}\text{2}$,$\text{1}\text{2}$}_n+1 as collocation points. Uniform convergence is shown to hold for the first iterate. Convergence is proved via Vainikko's embedding method, and stability is demonstrated by generalizing the approach of Atkinson for integral equations of the second kind.     

On a generalized X-ray transform and a new method for defect detection using the medium electronic density

We consider a generalization of the X-ray transform which maps a function $\text{f(}\text{V}\text{)}$ defined in $\text{R}{}^3$ onto its integrals over a pair of parallel straight lines and not over a single straight line as in a conventional X-ray transform. This new transformation arises from the image formation by double Compton-scattered radiation in transmission imaging. The problem of reconstructing $\text{f(}\text{V}\text{)}$ from its line pair integrals is formulated as an inverse problem for this generalized X-ray transform. Exploiting the line duality in the new transformation, we derive an equivalent nonlinear integral equation for $\text{f(}\text{V}\text{)}$. Special classes of solutions can be constructed. They may serve as basis for a new method of defect detection, using the medium electronic density, in non-destructive inspection. To cite this article: M.K. Nguyen, T.T. Truong, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Stopping times and an extension of stochastic integrals in the plane

Let (Ω,$\text{J}$,P;$\text{J}$_z) be a probability space with an increasing family of sub-σ-fields {$\text{J}$_z, z ∈ D}, where D = [0, ∞) × [0, ∞), satisfying the usual conditions. In this paper, the stochastic integral with respect to an $\text{J}$_z-adapted 2-parameter Brownian motion for integrand processes in the class $\text{C}$₂($\text{J}$_z) is extended, by means of truncations cations by {0, 1}-valued 2-parameter stopping times, to integrand processes that are $\text{J}$_z-adapted and continuous. The stochastic integral in the plane thus extended resembles a locally square integrable martingale in the 1-parameter setting. A definition of a parameter-space valued, i.e., D-valued, stopping time is also given and its characteristic process is related to a {0, 1}-valued 2-parameter stopping time.     

On a symptotic methods for Fredholm–Volterra integral equation of the second kind in contact problems

A method is used to obtain the general solution of Fredholm–Volterra integral equation of the second kind in the space $\text{L}{}_2\text{(Ω)×C(0,T),}\text{0⩽t⩽T<∞;Ω}$ is the domain of integrations.The kernel of the Fredholm integral term belong to $\text{C([Ω]×[Ω])}$ and has a singular term and a smooth term. The kernel of Volterra integral term is a positive continuous in the class C(0,T), while $\text{Ω}$ is the domain of integration with respect to the Fredholm integral term.Besides the separation method, the method of orthogonal polynomials has been used to obtain the solution of the Fredholm integral equation. The principal (singular) part of the kernel which corresponds to the selected domain of parameter variation is isolated. The unknown and known functions are expanded in a Chebyshev polynomial and an infinite algebraic system is obtained.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

Generalizations of magic graphs

A graph is magic if the edges are labeled with distinct nonnegative real numbers such that the sum of the labels incident to each vertex is the same. Given a graph finite G, an Abelian group $\text{g}$, and an element r(v) ∈ $\text{g}$ for every v ∈ V(G), necessary and sufficient conditions are given for the existence of edge labels from $\text{g}$ such that the sum of the labels incident to v is r(v). When there do exist labels, all possible labels are determined. The matroid structure of the labels is investigated when $\text{g}$ is an integral domain, and a dimensional structure results. Characterizations of several classes of graphs are given, namely, zero magic, semi-magic, and trivial magic graphs.     

Nonlinear perturbations of quasi-linear control systems

Several sufficient conditions are developed for controllability of the perturbed quasi-linear system $\text{x}\text{?}\text{= A(t, x, u)x + B(t, x, u)u + f(t, x, u)}$. In particular, these reduce to conditions on the perturbation function f which guarantee that if the linear system $\text{x}\text{?}\text{= A(t)x + B(t)u}$ is controllable, then the system $\text{x}\text{?}\text{= A(t)x + B(t)u + f(t, x, u)}$ is controllable. These conditions are growth conditions in (x, u) and are obtained by solving a system of nonlinear integral equations.     

When is an operator the integral of a given spectral measure?

For a closed densely defined operator T on a complex Hilbert space $\text{H}$ and a spectral measure E for $\text{H}$ of countable multiplicity q defined on a σ-algebra $\text{B}$ over an arbitrary space Λ we give three conceptually differing but equivalent answers to the question asked in the title of the paper (Theorem 1.5). We then study the simplifications which accrue when T is continuous or when q = 1 (Sect. 4). With the aid of these results we obtain necessary and sufficient conditions for T to be the integral of the spectral measure of a given group of unitary operators parametrized over a locally compact abelian group Γ (Sect. 5). Applying this result to the Hilbert space $\text{H}$ of functions which are L₂ with respect to Haar measure for Γ, we derive a generalization of Bochner's theorem on multiplication operators (Sect. 6). Some results on the multiplicity of indicator spectral measures over Γ are also obtained. When Γ = $\text{R}$ we easily deduce the classical theorem about the commutant of the associated self-adjoint operator (Sect. 7).     

Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space $\text{H}$ the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from $\text{H}$^q to $\text{H}$^p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for $\text{H}$^q is the “inflation” of a spectral measure for $\text{H}$. In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫(in0, 2π]e^?iθE(dθ) is a unitary operator on $\text{H}$ and if T is a bounded linear operator from $\text{H}$^q to $\text{H}$^q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process (Uⁿx)_?∞^∞ ?$\text{H}$^q (for each x = (xⁱ)_i^j ∈ $\text{H}$^q, Uⁿx = (Uⁿxⁱ)_i=1^q), then T is a spectral integral, ∫(_0,2π)]Φ(θ) E(dθ), and the Banach norm of T, |T|_B = ess sup |Φ(θ)|_B.     

Codimension one spheres in Rn with double tangent balls

In contrast to the situation in $\text{R}$³, where a 2-sphere with double tangent balls at each point must be tamely embedded in $\text{R}$³, there exist wild (n?1)-spheres in $\text{R}$ⁿ for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in $\text{R}$ⁿ. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.     

Invariant subspaces of certain subnormal operators

Let T be a subnormal, nonnormal operator on a Hilbert space and suppose that the point spectrum of $\text{T}{}^1$ is empty. Then there exist vectors x ≠ 0 for which $\text{(T}{}^1\text{? zI)}{}^{\mathrm{?1}}\text{x}$ exists and is weakly continuous for all z. It is shown that under certain conditions, the Cauchy integral of this vector function taken around an appropriate contour, not necessarily lying in the resolvent set of $\text{T}{}^1$, leads to a proper (nontrivial) invariant subspace of $\text{T}{}^1$.     

On the value of a stopped set function process

For certain types of stochastic processes {X_n | n ∈ $\text{N}$}, which are integrable and adapted to a nondecreasing sequence of σ-algebras $\text{F}$_n on a probability space (Ω, $\text{F}$, P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {$\text{F}$_n | n ∈ $\text{N}$}, when is$\text{sup}{}_{\mathrm{s}}\text{∫}{}_{\mathrm{\Omega }}\text{| X}{}_{\mathrm{\sigma }}\text{| dP}$finite, and when is there a stopping time $\text{σ}$ for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research.     

Effect of delays on functional differential equations

The perturbed functional differential equation $\text{x}\text{?}\text{(t) = L(x}{}_{\mathrm{t}}\text{) + h(x}{}_{\mathrm{t}}\text{)}$ is considered with the assumption that h is Lipschitzian in W_1,∞. Using integral manifold techniques, this equation is reduced to the equivalent ordinary differential equation $\text{u}\text{?}\text{= Bu + Ψ(0)h(ΛΦu)}$. A bifurcation problem is considered for the former equation. Illustrative examples are worked.     

Multiplicative partial integration and the Trotter Product Formula

For a long time it has been apparent that the Trotter Product Formula, a simple version of which reads $\text{exp}\text{{(A + B)t} =}\text{lim}{}_{\mathrm{n\to \infty }}\text{exp {(}\text{At}\text{n}\text{) · (}\text{Bt}\text{n}\text{)}}{}^{\mathrm{n}}$, where A, B are (non-commuting) operators and t is real, is related to the formula for the multipicative integral of the sum of two functions. But the precise connection between the two has not been exposed in the literature. Our purpose is to fill this lacuna. The theory of the multiplicative integral is outlined in Section 1, and the formula for the integral of the sum of two functions and the Trotter Product Formula are discussed in Section 2. Possible extensions are alluded to in Section 3.     

Explicit error terms for asymptotic expansions of Mellin convolutions

Explicit expressions are derived for the error terms associated with the asymptotic expansions of the convolution integral $\text{I(λ) = ∝}{}_0{}^{\mathrm{\infty }}\text{?(t) h(λt) dt}$, where h(t) and $\text{?(t)}$ are algebraically dominated at both 0⁺ and + ∞. Examples included are Fourier, Bessel, generalized Stieltjes, Hilbert and “potential” transforms.     

Periodic solutions for perturbed differential equations

A variant of the Alekseev variation of constants integral equation is obtained relating the solutions of systems of the form $\text{x}\text{?}\text{= f(t, x, λ)}$ and $\text{y}\text{?}\text{= f(t, y, ψ(t, y)) + g(t, y)}$. For the case when f, g, and ψ have period P in t several theorems are given for the existence of periodic solutions extending known results when f is linear in x and does not depend on the parameter m-vector λ. Comparison with an older technique gives hypotheses where the method above is advantageous for establishing periodic solutions. An example is given for constructing limit cycles of autonomous second-order systems.     

Univalent solutions of Briot-Bouquet differential equations

Let β and γ be complex numbers and let h(z) be regular in the unit disc U. This article studies the Briot-Bouquet differential equation $\text{q(z) + zq′(z)}\text{(βq(z) + γ)}\text{= h(z)}$. Sufficient conditions are obtained for both the regularity and univalency of the solution in U. In addition, applications of these results to differential subordinations, integral operators and univalent functions are given.     

Numerical solution of non-linear Volterra integral equations

This paper deals with non-linear Volterra integral equations of the type $\text{y(x)}\text{=}\text{f(x)}\text{+ ?}{}_0{}^{\mathrm{<mtext>x</mtext>}}\text{H[t, x, y (t), y (x)] dt}$. Convergence criteria are given (in the same sense of the maximum and C^a norms) for the numerical solution of this type of Volterra integral equation. Several numerical methods are compared.     

Propriétés de l'intégrale de Cauchy Harish-Chandra pour certaines paires duales d'algèbres de LieProperties of the Cauchy Harish-Chandra integral for some dual pairs of Lie algebras

We consider a symplectic group Sp and an reductive and irreductible dual pair (G,G′) in Sp in the sense of R. Howe. Let $\text{g}$ (resp. $\text{g}\text{′}$) be the Lie algebra of G (resp. G′). T. Przebinda has defined a map $\text{Chc}$, called the Cauchy Harish-Chandra integral from the space of smooth compactly supported functions of $\text{g}$ to the space of functions defined on the open set $\text{g}{}^{\mathrm{<mtext>\prime </mtext><mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>reg</mtext>}}$ of semisimple regular elements of $\text{g}\text{′}$. We prove that these functions are invariant integrals if G and G′ are linear groups and they behave locally like invariant integrals if G and G′ are unitary groups of same rank. In this last case, we obtain the jump relations up to a multiplicative constant which only depends on the dual pair. To cite this article: F. Bernon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 945–948.