Equiconvergence and equisummability of nonharmonic Fourier expansions with ordinary trigonometric series

Givenf ε L(?π, π), we consider its nonharmonic Fourier series \(f(x) \sim \sum c_n e^{i\lambda _n x} \) , where λ_n are the roots of the entire function L(z) = ∫ _-π ^π e ^izt dσ (t). We show that this series is equiconvergent, uniformly inside (-π, π), and equisummable with the Fourier series off with respect to the trigonometric system if σ′ (t) =k (t) (π - ∣t∣)^-α, α ε (0, 1), vark <∞, k (π ?0) ≠ 0,k (? π + 0) ≠ 0.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

On the Semicontinuity of Curvature Integrals

Let H_j(K, ·) be the j – th elementary symmetric function of the principal curvatures of a convex body K in Euclidean d – space. We show that the functionals ∫_bd f(H_j(K, x)) dℋ︁^d—1(x) depend upper semicontinuously on K, if f : [0, ∞) is concave, lim_t→0f(t) = 0, and lim_t→∞f(t)/t = 0. An analogous statement holds for integrals of elementary symmetric functions of the principal radii of curvature.     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

Remarks on the equation dXt = a(Xt)dBt

We prove that if a global solution of the equation dX_t = a(X_t) dB_t, X₀ = x exists for some x ? $\text{R}$ and ∫^∞₀a²(X_s)ds = ∞, then one must have a ≠ 0 a.e.     

The identification problem for the exponential Radon transform

The exponential Radon transform, which arises in single photon emission computed tomography, is defined by ? ?(μ:ω,s) = ∫_R?(sω + tomega;^?) e^μt dt?. Here ? is a compactly supported distribution in the plane which represents the location and intensity of a radio-pharmaceutical in a body of constant, but unknown, attenuation μ, and ω is a direction. The identification problem is to determine the attenuation μ from the data ?? with ? unknown. We will show that μ can be determined from the data if and only if ? is not a radial distribution and give formulae for computing μ when ? is not radial.     

The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data

Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation u_t = u_xx + p(t)u + φ, in [0,1] × (0,T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ∫₀¹k(x)u(x,t)dx = E(t), 0 ≤ t ≤ T, for known functions E(t), k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ∫₀^s(t) u(x,t)dx = E(t), 0 < t ≤ T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On Certain Fejér and Tchebychev Type Integrals

Let m and n denote a pair of positive integers. In this paper, we call upon the Hadamard product and computer algebra techniques to evaluate the Fejér integral π ^?1 ∫₀ ^π (sin mθ / sin θ) ²ⁿ dθ. Using symmetry arguments, it is proved that the value of this integral is an odd polynomial in m of degree 2n ? 1. This permits using polynomial curve fitting methods and mathematical software packages to obtain evaluation formulas for n relatively small. Some cases of the above integral with 2n replaced by 2n + 1 are also discussed. A familiar identity shows that these yield evaluations of integrals of powers of certain Tchebychev polynomials.     

Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·)

We study the Riesz potentials I_αf on the generalized Lebesgue spaces L^p(·)(?^d), where 0 < α < d and I_αf(x) ? ∫equation/tex2gif-inf-3.gif |f(y)| |x – y|^{α – d} dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |x – y|) and is constant outside some large ball, we prove that I_α : L^p(·)(?^d) → L^p?(·)(?^d), where . If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to on ?^d such that there exists a bounded linear extension operator ? : W^1,p(·)(Ω) ? (?^d), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings W^k,p(·)(?^d) ?L^p*(·)(R^d) with and W^1,p(·)(Ω) ? L^p*(·)(Ω) for k = 1. We show compactness of the embeddings W^1,p(·)(Ω) ? L^q(·)(Ω), whenever q(x) ≤ p*(x) – ε for some ε > 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Evolution of the Weyl and Maxwell Fields in Curved Space—Times

The covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (u, φ) of a space-time (M, g) are considered. It is shown that the condition g⁰⁰(x) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δ_tφ = L_(s)φ. Here L_(s) is a linear first order differential operator on the pre—Hilbert space (C (U_t, ^2s+1). (…)), where U_t ? IR³ is the image of the coordinate map of the spacelike hyper-surface t = const, and (φ, C) = ?U_t ? *Q d⁽³⁾x with a suitable Hermitian n × n- matrix Q = Q(t,x). The total energy of the spinor field ? with respect to U_t is then simply given by E = 〈?,?〉. In this way inequalities for the energy change rate with respect to time, δ_t|?|² = 2Re (?, L_(s)?) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.     

On a nonlinear stochastic integral equation with application to control systems

The aim of the present paper is to study a nonlinear stochastic integral equation of the form

Evolution Semigroups and Product Formulas for Nonautonomous Cauchy Problems

In this paper, we study nonautonomous Cauchy problems (NCP) {^{u̇(t) = A(t)u(t)}_{u(s) = x ∈ X} for a family of linear operators (A(t))_t∈I on some Banach space X by means of evolution semigroups. In particular, we characterize “stability” in the so called “hyperbolic case” on the level of evolution semigroups and derive a product formula for the solutions of (NCP). Moreover, in Section 4 we connect the “hyperbolic” and the “parabolic” case by showing, that integrals ∫^t_s A(τ) dτ always define generators. This yields another product formula.     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

Numerical evaluation of finite Fourier integrals

A computationally efficient algorithm for evaluating Fourier integrals ∫¹_?1?(x)e^iωxdx using interpolatory quadrature formulas on any set of collocation points is presented. Examples are given to illustrate the performances of interpolatory formulas which are based on the applications of the Fejér, Clenshaw—Curtis, Basu and the Newton—Cotes points. Initially, the formulas for nonoscillatory integrals are generated and then generalizations to finite Fourier integrals are made. Extensions of this algorithm to some other weighted integrals are also considered.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

On nonautonomous linear systems of differential and difference equations with -symmetric coefficient matrices

Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e^-2πi/k, d₀+d₁++d_k-1=n, and with . We say that is R-symmetric if R(t)A(t)R^-1(t)=A(t), , and we show that if A is R-symmetric then solving x^′=A(t)x or x^′=A(t)x+f(t) reduces to solving k independent d_ℓ×d_ℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations.     

Differentiation formulas for stochastic integrals in the plane

For a one-parameter process of the form X_t=X₀+∫^t₀φ_sdW_s+∫^t₀ψ_sds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(X_t) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W₂, z∈R²₊} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let X_z be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X_z) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X_z) on increasing paths in $\text{R}{}^2{}_{\mathrm{+}}$.     

Some results about the cross-correlation function between two maximal linear sequences

Let {a₁} and {a_d1} be two maximal linear sequences of period pⁿ ? 1. The cross-correlation function is defined by C_d(t) =

for t = 0, t…pⁿ ? 2, where $\text{ζ = exp(2π}\text{1}\text{p}\text{)}$. We find some new general results about C_d(t). We also determine the values and the number of occurences of each value of C_d(t) for several new values of d.     

Excursion theory for rotation invariant Markov processes

Summary Let (X _t,P ^x) be a rotation invariant (RI) strong Markov process onR ^d{0} having a skew product representation [|X _t |, ], where ( _t ) is a time homogeneous, RI strong Markov process onS ^d–1, |X _t|, and _t are independent underP ^x andA _t is a continuous additive functional of |X _t|. We characterize the rotation invariant extensions of (X _t,P ^x) toR ^d. Two examples are given: the diffusion case, where especially the Walsh's Brownian motion (Brownian hedgehog) is considered, and the case where (X _t,P ^x) is self-similar.     

An inversion formula for a class of integral transforms,II

In this note we give a procedure for inverting the integral transform f(x) = ∫₀^∞k(xt) φ(t) dt, where the functions f(x) and k(x) are known and φ(x) is to be found. The inversion is accomplished in two steps: by first defining a transforming function, which is an integral, followed by the application of an infinite order differential operator.