Laplace transform inversion and padé-type approximants

A method for the numerical inversion of the Laplace transform of a functions f is to approximate it by rational functions f_m(z), and then to use the inverse transforms F_m(t) of f_m(z) as approximation of the inverse transform F(t) of f(z).As in Tricomi's method we define f_m(z) as a partial sum of a series expansion, which is also a Padé-type approximant to f with one pole. Then F_m(t) is the partial sum of the expansion of F(t) in terms of Laguerre polynomials.We prove mean square and uniform convergence results. The study for the choice of the pole of f_m is used to define a best Padé-type approximant with one pole. It permits the use of the method of inversion by Laguerre polynomials, with good numerical results for functions having essential singularities.     

Modified Dyadic Integral and Fractional Derivative on ℝ+

For functions from the Lebesgue space L(?₊), we introduce the modified strong dyadic integral J _α and the fractional derivative D ^(α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(?₊). We find a countable set of eigenfunctions of the operators D ^(α) and J _α, α > 0. We also prove the relations D ^(α)(J _α(f)) = f and J _α(D ^(α)(f)) = f under the condition that $\smallint _{\mathbb{R}_ + } f(x)dx = 0$ . We show the unboundedness of the linear operator $J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$ , where L _{J α} is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$ . Moreover, for a function f ∈ L(?₊) and a given point x ∈ ?₊, we introduce the modified dyadic derivative d ^(α)(f)(x) and the modified dyadic integral j _α(f)(x). We prove the relations d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) at each dyadic Lebesgue point of the function f.     

The inverse laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation

When the Laplace transform F(p) of a function f(x) has no poles but is singular only on the real negative semiaxis because of a cut required to make it single-valued, the inverse transform f(x) can easily be computed by means of the integral of a real-valued function. This result is applied to the calculation of a class of exact eternal solutions of the Boltzmann equation, recently found by the authors. The new approach makes it easier to prove that these solutions are positive, as well as to study their asymptotics.     

On R-robustly entropy-expansive diffeomorphisms

For a homoclinic class H(p _f ) of f ?? Diff¹(M), f?OH(p _f ) is called R-robustly entropy-expansive if for g in a locally residual subset around f, the set ?? _? (x) = {y ?? M: dist(g ⁿ (x), g ⁿ (y)) ?? g3 (?n ?? ?)} has zero topological entropy for each x ?? H(p _g ). We prove that there exists an open and dense set around f such that for every g in it, H(p _g ) admits a dominated splitting of the form E ?? F ₁ ?? ... ?? F _k ?? G where all of F _i are one-dimensional and non-hyperbolic, which extends a result of Pacifico and Vieitez for robustly entropy-expansive diffeomorphisms. Some relevant consequences are also shown.     

Graham’s pebbling conjecture on products of many cycles

A pebbling move on a connected graph G consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define f_t(G) as the smallest number such that whenever f_t(G) pebbles are on G, we can move t pebbles to any specified, but arbitrary vertex. Graham conjectured that f₁(G×H)≤f₁(G)f₁(H) for any connected G and H. We define the α-pebbling number α(G) and prove that α(C_pj×?×C_p2×C_p1×G)≤α(C_pj)?α(C_p2)α(C_p1)α(G) when none of the cycles is C₅, and G satisfies one more criterion. We also apply this result with G=C₅×C₅ by showing that C₅×C₅ satisfies Chung’s two-pebbling property, and establishing bounds for f_t(C₅×C₅).     

Nielsen periodic point theory for periodic maps on orientable surfaces

Let denote a periodic self map of minimal period m on the orientable surface of genus g with g>1. We study the calculation of the Nielsen periodic numbers NPn(f) and NΦn(f). Unlike the general situation of arbitrary maps on such surfaces, strong geometric results of Jiang and Guo allow for straightforward calculations when n≠m. However, determining NPm(f) involves some surprises. Because fm=idFg, fm has one Nielsen class Em. This class is essential because L(idFg)=χ(Fg)=2−2g≠0. If there exists k<m with L(fk)≠0 then Em reduces to the essential fixed points of fk. There are maps g (we call them minLef maps) for which L(gk)=0 for all k<m. We show that the period of any minLef map must always divide 2g−2. We prove that for such maps Em reduces algebraically iff it reduces geometrically. This result eliminates one of the most difficult problems in calculating the Nielsen periodic point numbers and gives a complete trichotomy (non-minLef, reducible minLef, and irreducible minLef) for periodic maps on Fg.We prove that reducible minLef maps must have even period. For each of the three types of periodic maps we exhibit an example f and calculate both NPn(f) and NΦn(f) for all n. The example of an irreducible minLef map is on F₄ and is of maximal period 6. The example of a non-minLef map is on F₂ and has maximal period 12 on F₂. It is defined geometrically by Wang, and we provide the induced homomorphism and analyze it. The example of an irreducible minLef map is a map of period 6 on F₄ defined by Yang. No algebraic analysis is necessary to prove that this last example is an irreducible minLef map. We explore the algebra involved because it is intriguing in its own right. The examples of reducible minLef maps are simple inversions, which can be applied to any Fg. Using these examples we disprove the conjecture from the conclusion of our previous paper.     

The extended gamma class and applications

The gamma class Γ _α (g) consists of positive and measurable functions that satisfy f(x+yg(x))/f(x)→exp(αy). In most cases, the auxiliary function g is Beurling varying, i.e. g(x)/x→0 and g∈Γ₀(g). Taking h=logf, we find that h∈EΓ _α (g,1), where EΓ _α (g,a) is the class of ultimately positive and measurable functions that satisfy (f(x+yg(x))?f(x))/a(x)→αy. In this paper, we discuss local uniform convergence for functions in the classes Γ _α (g) and EΓ _α (g,a). From this we obtain several representation theorems. We also prove some higher order relations for functions in the classes Γ _α (g) and EΓ _α (g,a). Some applications conclude the paper.     

Arithmetical properties of linear recurrent sequences

Let F(z)∈R[z] be a polynomial with positive leading coefficient, and let α>1 be an algebraic number. For r=degF>0, assuming that at least one coefficient of F lies outside the field Q(α) if α is a Pisot number, we prove that the difference between the largest and the smallest limit points of the sequence of fractional parts {F(n)αn}_n=1,2,3,… is at least 1/?(Pr+1), where ? stands for the so-called reduced length of a polynomial.     

Construction of bent functions via Niho power functions

A Boolean function with an even number n=2k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f(x)=tr(α₁x^d₁+α₂x^d₂), α₁,α₂,x∈F_2ⁿ, are considered, where the exponents d_i (i=1,2) are of Niho type, i.e. the restriction of x^d_i on F_2^k is linear. We prove for several pairs of (d₁,d₂) that f is a bent function, when α₁ and α₂ fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F_2ⁿ are bijective.     

An improvement of Montel's criterion

Let F be a family of holomorphic functions in a domain D, and let a(z), b(z) be two holomorphic functions in D such that a(z)?b(z), and a(z)?a^′(z) or b(z)?b^′(z). In this paper, we prove that: if, for each f∈F, f(z)−a(z) and f(z)−b(z) have no common zeros, f^′(z)=a(z) whenever f(z)=a(z), and f^′(z)=b(z) whenever f(z)=b(z) in D, then F is normal in D. This result improves and generalizes the classical Montel's normality criterion, and the related results of Pang, Fang and the first author. Some examples are given to show the sharpness of our result.     

Minimal periods of semilinear evolution equations with Lipschitz nonlinearity

It is known that any periodic orbit of a Lipschitz ordinary differential equation must have period at least 2π/L, where L is the Lipschitz constant of f. In this paper, we prove a similar result for the semilinear evolution equation du/dt=-Au+f(u): for each α with 0?α?1/2 there exists a constant K_α such that if L is the Lipschitz constant of f as a map from D(A^α) into H then any periodic orbit has period at least K_αL^-1/(1-α). As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier-Stokes equations with periodic boundary conditions.     

Annihilating sets for the short time Fourier transform

We obtain a class of subsets of R2d such that the support of the short time Fourier transform (STFT) of a signal f∈L²(Rd) with respect to a window g∈L²(Rd) cannot belong to this class unless f or g is identically zero. Moreover we prove that the L²-norm of the STFT is essentially concentrated in the complement of such a set. A generalization to other Hilbert spaces of functions or distributions is also provided. To this aim we obtain some results on compactness of localization operators acting on weighted modulation Hilbert spaces.     

Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

Tauberian and Abelian theorems for rapidly decaying distributions and their applications to stable laws

We establish some assertions of Tauberian and Abelian types which enable us to find connections between the asymptotic properties of the Laplace transform at infinity and the asymptotics of the corresponding densities of rapidly decaying distributions (at infinity or in some neighborhood of zero). As applications of our Tauberian type theorems we present asymptotics for the density f ^(α,ρ)(x) of “extreme” stable laws with parameters (α, ρ) for ρ = ±1 and x lying in the domain of rapid decay of f ^(α,ρ)(x). This asymptotics had been found in [1–5] by a more complicated method.     

Weighted Berezin transform in the polydisc

For c>−1, let νc denote a weighted radial measure on C normalized so that νc(D)=1. If f is harmonic and integrable with respect to νc over the open unit disc D, then for every ψ∈Aut(D). Equivalently f is invariant under the weighted Berezin transform; Bcf=f. Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any 1?p<∞ and c₁,c₂>−1, a function f∈Lp(D²,νc₁×νc₂) which is invariant under the weighted Berezin transform; Bc₁,c₂f=f needs not be 2-harmonic.     

Algorithm for the computation of Bessel function integrals

A fortran subroutine is given for the computation of integrals of the form ∫^c₀f(x)J_v(αx)dx, where v = 0, 1,…,10.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Uniformly bounded composition operators in the banach space of bounded (p, k)-variation in the sense of Riesz-Popoviciu

We prove that if the composition operator F generated by a function f: [a, b] × ? → ? maps the space of bounded (p, k)-variation in the sense of Riesz-Popoviciu, p ≥ 1, k an integer, denoted by RV_(p,k)[a, b], into itself and is uniformly bounded then RV_(p,k)[a, b] satisfies the Matkowski condition.