We identify measures arising in the representations of products of generalized Stieltjes transforms as generalized Stieltjes transforms, provide optimal estimates for the size of those measures, and address a similar issue for generalized Cauchy transforms. In the latter case, in two particular settings, we give criteria ensuring that the measures are positive. On this way, we also obtain new, applicable conditions for representability of functions as generalized Stieltjes transforms, thus providing a partial answer to a problem posed by Sokal and shedding a light at spectral multipliers emerged recently in probabilistic studies. As a byproduct of our approach, we improve several known results on Stieltjes and Hilbert transforms.
For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the…
In this paper we apply generalized Stieltjes transform representation to study the generalized hypergeometric function. Among the results thus proved are new integral representations, inequalities, properties of the Padé table and the properties of the generalized hypergeometric function as a conformal map.
Nondegenerate truncated indefinite Stieltjes moment problem in the class $\mathbf{N}_κ^{k}$ of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step-by-step algorythm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recently in \cite{DK15}. Explicit formula for the resolvent matrix in terms of generalized Stieltjes polynomials is found.
In this paper, new sharp weighted generalizations of Ostrowski and generalized trapezoid type inequalities for the Riemann--Stieltjes integrals are proved. Several related inequalities are deduced and investigated. New Simpson's type inequalities for $\mathcal{RS}$--integral are pointed out. Finally, as application; an error estimation of a general quadrature rule for $\mathcal{RS}$--integral via Ostrowski--generalized trapezoid quadrature formula is given.
In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related to the generalized Stieltjes constants, and some integrals involving the function $x\mapsto \ln \ln(1/x)$.
We obtain a characterization of generalized Stieltjes functions of any order λ> 0 in terms of inequalities for their derivatives on (0,\infty). When λ=1, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.
It is shown that a function $f$ is a generalized Stieltjes function of order $λ>0$ if and only if $x^{1-λ}(x^{λ-1+k}f(x))^{(k)}$ is completely monotonic for all $k\geq 0$, thereby complementing a result due to Sokal. Furthermore, a characterization of those completely monotonic functions $f$ for which $x^{1-λ}(x^{λ-1+k}f(x))^{(k)}$ is completely monotonic for all $k\leq n$ is obtained in terms of properties of the representing measure of $f$.
The paper surveys the basic properties of generalized Stieltjes functions including some new ones. We introduce the notion of the exact Stieltjes order and give a criterion of exactness, simple sufficient conditions and some prototypical examples. The paper includes an appendix, where we define the left sided Riemann-Liouville and the right sided Kober-Erdelyi fractional integrals of measures supported on half axis and give inversion formulas for them.
For each of the eight $n$-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential operator (for two sets of parameters): a so-called transmutation property. This leads to eight fractional integration formulas and four generalized Stieltjes transform formulas for each of the six different explicit solutions of the hypergeometric differential equation, by letting the transforms act on …
We consider mapping properties of the iterated Stieltjes transform, establishing its new relations with the iterated Hilbert transform (a singular integral) on the half-axis and proving the corresponding convolution and Titchmarsh's type theorems. Moreover, the obtained convolution method is applied to solve a new class of singular integral equations.
A class of Stieltjes functions of finite type is introduced. These satisfy Widder's conditions on the successive derivatives up to some finite order, and are not necessarily smooth. We show that such functions have a unique integral representation, along some generic kernel which is a truncated Laurent series approximating the standard Stieltjes kernel. We then obtain a two-to-one correspondence, via the logarithmic derivative, between these functions and a subclass of hyperbolically monotone functions of finite type. This correspondence generalizes a representation of HCM functions in terms o…
Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are generalised to entire functions of genus $p$ with negative zeros. Derivatives of order comparable to $p$ of the expression above are related to generalised Stieltjes functions of order $p+1$. Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler's gamma function on vertical lines can be shar…
The aim of this paper is to investigate in detail the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.
We present a elementary approach to asymptotic behavior of generalized functions in the Cesàro sense. Our approach is based on Yosida's subspace of Mikusiński operators. Applications to Laplace and Stieltjes transforms are given.
Maximal angular operator sends a function defined in a sector of the complex plane to a Maximal angular operator sends a function defined in a sector of the complex plane with vertex at 0 to the function of modulus obtained by maximizing over argument. Compositions of the so defined maximal angular operator (in suitable sectors) with the Poisson, Stieltjes and Laplace transforms are shown to be bounded (nonlinear) operators from $L^p$ to $L^q$ for the same values of $p$ and $q$ as their standard counterparts.
We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces $L_w^{p(\cdot)}(0,\infty)$, assuming that the exponent function $p(\cdot)$ is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals $\{ (0,b) : b>0\}$ on $(0,\infty)$. Our results extend those in \cite{DMRO1} for the constant exponent $L^p$ spaces with weights. We also give two applications: the first is a weighted version of Hilbert's in…
In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial growth. We also allow the Fourier transform to exist in the principal value sense. A function is called regulated if it has a left limit and a right limit at each point. The main inversion theorem is obtained by solving the differential equation $df(t)-iωf(t)=g(t)$ for a regulated function $f$, where $ω$ is a complex number with positive imaginary part. This …
In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the o…
In our previous work we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences of these properties. In particular, we find new integral representations of the generalized hypergeometric functions, evaluate a number of integrals of their products, compute the jump and the average value of the the generalized hypergeometric function over the branch cut, establish new inequalities for this function in the half plane Re(z)<1. Furthermore,…
Pleijel's inequality is an approximate inversion formula for the Stieltjes transform (or Cauchy integral) of a distribution function on positive semi-axis. It implies a Tauberian theorem due to Malliavin. The proposed analogs of Pleijel's inequality deal with (1) approximate recovery of the Riesz means of the distribution function from its Stieltjes transform, and (2) approximate recovery of the distribution function with power growth for which the ordinary Stieltjes transform does not exist. In the latter case, a power of the Cauchy (or Stieltjes) kernel is used to define the "generalized Sti…
By using structural and asymptotic properties of the Kontorovich-Lebedev transform associated with Minkowski's question mark function, we give an affirmative answer to the question posed by R. Salem (Trans. Amer. Math. Soc., 53 (3), (1943), p. 439) whether its Fourier-Stieltjes transform vanishes at infinity. This paper is substituted with a correct alternative and affirmative solution of Salem's problem and its generalization for derivatives of any order of the Fourier-Stieltjes transform of the Minkowski'i question mark function. The question is finally solved.
We present the exact solution for a set of nonlinear algebraic equations $\frac{1}{z_l}= πd + \frac{2 d}{n} \sum_{m \neq l} \frac{1}{z_l-z_m}$. These were encountered by us in a recent study of the low energy spectrum of the Heisenberg ferromagnetic chain \cite{dhar}. These equations are low $d$ (density) ``degenerations'' of more complicated transcendental equation of Bethe's Ansatz for a ferromagnet, but are interesting in themselves. They generalize, through a single parameter, the equations of Stieltjes, $x_l = \sum_{m \neq l} 1/(x_l-x_m)$, familiar from Random Matrix theory. It is shown t…
We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy-Khinchin representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.
Let dμ(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of dμ(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present paper, we consider a matrix interpretation of the unwrapping transformation from S(λ) to λS(λ^2), which is intimately related to the simplest case of polynomial mappings. More precisely, it is shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to th…
We construct certain Rajchman measures by using integrability properties of the Fourier and Fourier-Stieltjes transforms. Further we show that Minkowski's question mark function ?(x), which is a singular monotone function, belongs to one of these classes. This circumstance leads us to a rigorous proof of the problem posed by R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943), p. 439) whether Fourier-Stieltjes coefficients of this function vanish at infinity and whose affirmative solution was recently announced by the author in [13]. Moreover, we generalize this problem, proving that derivatives o…
We find an integral representation for the generalized hypergeometric function unifying known representations via generalized Stieltjes, Laplace and cosine Fourier transforms. Using positivity conditions for the weight in this representation we establish various new facts regarding generalized hypergeometric functions, including complete monotonicity, log-convexity in upper parameters, monotonicity of ratios and new proofs of Luke's bounds. Besides, we derive two-sided inequalities for the Bessel type hypergeometric functions by using their series representations.
Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $\int_0^a f(x)/(ω+ x)^ρ \, \mathrm{d}x$ under the assumption that the extension of $f(x)$ in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of $f(x)$. Finite par…
We construct certain Rajchman measures by using integrability properties of the Fourier and Fourier-Stieltjes transforms. In particular, we state a problem and prove that it is equivalent to the known and still unsolved question posed by R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943), p. 439) whether Fourier-Stieltjes coefficients of the Minkowski's question mark function vanish at infinity.
Employing the generalized Parseval equality for the Mellin transform and elementary trigonometric formulas, the iterated Hartley transform on the nonnegative half-axis (the iterated half-Hartley transform) is investigated in L_2. Mapping and inversion properties are discussed, its relationship with the iterated Stieltjes transform is established. Various compositions with the Fourier cosine and sine transforms are obtained. The results are applied to the uniqueness and universality of the closed form solutions for certain new singular integral and integro-functional equations.
The generalized Stieltjes--Wigert polynomials depending on parameters 0\le p<1 and 0<q<1 are discussed. By removing the mass at zero of the N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization, we obtain a discrete measure μ^M which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials (P^M_n). As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well…
We investigate Laplace type and Laplace-Stieltjes type multipliers in the $d$-dimensional setting of the Dunkl harmonic oscillator with the associated group of reflections isomorphic to $\mathbb{Z}_2^d$ and in the related context of Laguerre function expansions of convolution type. We use Calderón-Zygmund theory to prove that these multiplier operators are bounded on weighted $L^p$, $1<p<\infty$, and from $L^1$ to weak $L^1$.
Let h be a real-analytic function in the neighborhood of some compact set K on the plane. We show that for any complex measure on the Euclidean space of a finite total variation without singular components with the Fourier--Stieltjes transform f(y) there exists another measure of a finite total variation with the Fourier transform g(y) with the property g(y)=h(f(y)) for each y such that f(y) belongs to K.
We find two-sided inequalities for the generalized hypergeometric function of the form ${_{q+1}}F_{q}(-x)$ with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of ${_{q+1}}F_{q}(-x)$ at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. T…
