arXiv subjects

S. Yakubovich

Publications and source records attributed to S. Yakubovich.

7 recordsLinked to original sources

arXiv2026

A proof of Askey's convexity conjecture

For $-1<α\leq1/2$, let $J_α$ be the Bessel function of the first kind and let $j_{α,2}$ be its second positive zero. Define $β(α)<α+1$ by $$ \int_0^{j_{α,2}}u^{-β(α)}J_α(u)\,du=0. $$ We prove that $β''(α)>0$ for $-1<α\leq1/2$, including the left second derivative at $α=1/2$. The continuous extension $β(-1)=0$ is strictly convex on $[-1,1/2]$, strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.

math.CA↗

arXiv2022

Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals

Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function $K_{iτ}(x)$. The results can be applied, for instance, to study the summability of the divergent Kontorovich-Lebedev integrals in the sense of Jones. Namely, we answer affirmatively a question (cf. [6]) whether these integrals converge for even entire functions of the exponential type in a weak sense.

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arXiv2013

On the iterated Stieltjes transform and its convolution with applications to singular integral equations

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.

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arXiv2012

The Kontorovich-Lebedev transform as a map between $d$-orthogonal polynomials

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this $KL_α$-transform over certain polynomial sequences will be under discussion, and a special attention will be given the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the $KL_α$-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose $KL_α$-transform is a $d$-orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and $d$ is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.

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arXiv2012

Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.

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arXiv2011

On a nonorthogonal polynomial sequence associated with Bessel operator

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them, its explicit expression, the connection with the Euler numbers, its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality, it is possible to associate to the canonical element of its dual sequence a positive-definite measure as long as certain stronger constraints are imposed.

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arXiv2007

Hardy-type theorem for functions orthogonal with respect to their zeros. The Jacobi weight case

Motivated by G. H. Hardy's 1939 results \cite{Hardy} on functions orthogonal with respect to their real zeros $λ_{n}, n=1,2,... $, we will consider, within the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval $(0,1)$, that is, the functions $f(z)=z^νF(z), ν\in \mathbb{R}$, where $F$ is entire and \begin{equation*} \int_{0}^{1}f(λ_{n}t)f(λ_{m}t)t^α(1-t)^βdt=0,\quad α>-1-2ν, β>-1, \end{equation*}% when $n\neq m$. Considering all possible functions on this class we are lead to the discovery of a new family of generalized Bessel functions including Bessel and Hyperbessel functions as special cases.

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