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We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.
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A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the Legendre-Fenchel transform. Motivated by the Hilbert space version of this result and by the well-known result saying that this convex conjugation transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the above-mentioned class of functions. It turns out that this nonlinear equation is very sensitive to the involved parameters and can have no solution, a unique solution, or several (possibly infinitely many) ones. Our analysis yields a few by-products, such as results related to positive definite operators, and to functional equations and inclusions involving monotone operators.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying that there exists a constant $p_0in(0,p_-)$, where $p_-:=mathop{mathrm {ess,inf}}_{xin mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(cdot)/p_0}(mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(cdot)}(mathbb R^n)$ via the first order Riesz transforms when $p_-in (frac{n-1}n,infty)$, and via compositions of all the first order Riesz transforms when $p_-in(0,frac{n-1}n)$.
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $Psi$-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.
In this work we verify the sufficiency of a Jensens necessary and sufficient condition for a class of genus 0 or 1 entire functions to have only real zeros. They are Fourier transforms of even, positive, indefinitely differentiable, and very fast decreasing functions. We also apply our result to several important special functions in mathematics, such as modified Bessel function $K_{iz}(a), a>0$ as a function of variable $z$, Riemann Xi function $Xi(z)$, and character Xi function $Xi(z;chi)$ when $chi$ is a real primitive non-principal character satisfying $varphi(u;chi)ge0$ on the real line, we prove these entire functions have only real zeros.
In this paper we investigate Lp-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.
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