arXiv subjects

Stefan Kahler

Publications and source records attributed to Stefan Kahler.

7 recordsLinked to original sources

arXiv2026

On Turán's inequality: new general criteria, nonnegative representations and the class of generalized Chebyshev polynomials

Originally, Turán's inequality states that if $(P_n(x))_{n\in\mathbb{N}_0}$ is the sequence of Legendre polynomials, then $Δ_n(x):=P_n^2(x)-P_{n+1}(x)P_{n-1}(x)\geq0$ for all $n\in\mathbb{N}$ and $x\in[-1,1]$. Gasper specified the parameters $α,β>-1$ for which the Jacobi polynomials $(R_n^{(α,β)}(x))_{n\in\mathbb{N}_0}$ satisfy Turán's inequality. Frequently, such results rely on the specific structure of the concrete orthogonal polynomials under consideration. Therefore, special focus has been put on general criteria (whose importance was particularly emphasized by Nevai). We provide two general criteria for Turán's inequality in terms of the three-term recurrence relation and also deal with sharper estimations of the Turán determinants $Δ_n(x)$. They extend earlier results of Szwarc and Berg--Szwarc. Applying our criteria to the class of generalized Chebyshev polynomials $(T_n^{(α,β)}(x))_{n\in\mathbb{N}_0}$, which are the quadratic transformations of the Jacobi polynomials, we find the companion to Gasper's above-mentioned result. At this stage, we also obtain nonnegative representations of $Δ_n(x)$. Finally, we study $2$-sieved polynomials and discuss further examples.

math.CA↗

arXiv2026

A new class of orthogonal polynomials

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations of the form $P_0(x)=1$, $P_1(x)=x$ and $x P_n(x)=a_n P_{n+1}(x)+c_n P_{n-1}(x)\;(n\in\mathbb{N})$, where $a_n$ and $c_n$ are positive and sum up to $1$. $(P_n(x))_{n\in\mathbb{N}_0}$ is said to satisfy nonnegative linearization of products if the product of any two polynomials $P_m(x)$, $P_n(x)$ is a convex combination of $P_{|m-n|}(x),\ldots,P_{m+n}(x)$. This property gives rise to a hypergroup structure and a sophisticated harmonic analysis. We are interested in examples such that both the original sequence $(P_n(x))_{n\in\mathbb{N}_0}$ and the sequence $(\widetilde{P_n}(x))_{n\in\mathbb{N}_0}$ which corresponds to switched roles of $(a_n)_{n\in\mathbb{N}}$ and $(c_n)_{n\in\mathbb{N}}$ satisfy nonnegative linearization of products. Such considerations were recently started by Lasser and Obermaier and can be motivated from a harmonic analytic, combinatorial or probabilistic point of view. However, Lasser and Obermaier left open the question whether examples besides the trivial example of the Chebyshev polynomials of the first kind $(T_n(x))_{n\in\mathbb{N}_0}$ (with $a_n\equiv c_n\equiv1/2$) actually exist. We provide a sufficient criterion and explicitly construct such nontrivial examples. Moreover, we provide characterizations of $(T_n(x))_{n\in\mathbb{N}_0}$ by additionally involving properties of the duals and Haar measures. Our criterion also enables us to solve open problems concerning the Haar measure of polynomial hypergroups stated by Kahler and Szwarc.

math.CA↗

arXiv2023

Dual spaces vs. Haar measures of polynomial hypergroups

Many symmetric orthogonal polynomials $(P_n(x))_{n\in\mathbb{N}_0}$ induce a hypergroup structure on $\mathbb{N}_0$. The Haar measure is the counting measure weighted with $h(n):=1/\int_\mathbb{R}\!P_n^2(x)\,\mathrm{d}μ(x)\geq1$, where $μ$ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property $h(n)\geq2\;(n\in\mathbb{N})$. We give sufficient criteria and particularly show that $h(n)\geq2\;(n\in\mathbb{N})$ if the (Hermitian) dual space $\widehat{\mathbb{N}_0}$ equals the full interval $[-1,1]$, which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the harmonic and functional analysis resulting from such expansions). Moreover, we construct two example types with $h(1)<2$. To our knowledge, these are the first such examples. The first type is based on Karlin-McGregor polynomials, and $\widehat{\mathbb{N}_0}$ consists of two intervals and can be chosen "maximal" in some sense; $h$ is of quadratic growth. The second type relies on certain compact operators; $h$ grows exponentially, and $\widehat{\mathbb{N}_0}$ is discrete.

math.CA↗

arXiv2023

Expansions and Characterizations of Sieved Random Walk Polynomials

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$ $n\in\mathbb{N}$ with $(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$. For every $k\in\mathbb{N}$, the $k$-sieved polynomials $(P_n(x;k))_{n\in\mathbb{N}_0}$ arise from the recurrence coefficients $c(n;k):=c_{n/k}$ if $k|n$ and $c(n;k):=1/2$ otherwise. A main objective of this paper is to study expansions in the Chebyshev basis $\{T_n(x)\colon n\in\mathbb{N}_0\}$. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version $\mathrm{D}_k$ of the Askey-Wilson operator $\mathcal{D}_q$. It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative and obtained from $\mathcal{D}_q$ by letting $q$ approach a $k$-th root of unity. However, for $k\geq2$ the new operator $\mathrm{D}_k$ on $\mathbb{R}[x]$ has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for $k$-sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator $\mathrm{A}_k$.

math.CA↗

arXiv2022

Harmonic analysis of little $q$-Legendre polynomials

Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as $L^1$-algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little $q$-Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their $L^1$-algebras have the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these $L^1$-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any $L^1$-algebra of a locally compact group; in the polynomial hypergroup context, weak amenability is rarely satisfied and of particular interest because it corresponds to a certain property of the derivatives of the underlying polynomials and their (Fourier) expansions w.r.t. the polynomial basis. To our knowledge, the little $q$-Legendre polynomials yield the first example of a polynomial hypergroup whose $L^1$-algebra is weakly amenable and right character amenable but fails to be amenable. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on the Fourier transformation on hypergroups, the Plancherel isomorphism, continued fractions, character estimations and asymptotic behavior.

math.FA↗

arXiv2020

Turán's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by $L^1$-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated $L^1$-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties holds (which is very different to $L^1$-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Turán's inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Turán type inequalities and transformations into more convenient orthogonal polynomial systems.

math.FA↗

arXiv2019

Nonnegative and strictly positive linearization of Jacobi and generalized Chebyshev polynomials

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $(P_n(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products, i.e., the product of any two $P_m(x),P_n(x)$ is a conical combination of the polynomials $P_{|m-n|}(x),\ldots,P_{m+n}(x)$. Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. In 1970, G. Gasper was able to determine the set $V$ of all pairs $(α,β)\in(-1,\infty)^2$ for which the corresponding Jacobi polynomials $(R_n^{(α,β)}(x))_{n\in\mathbb{N}_0}$, normalized by $R_n^{(α,β)}(1)\equiv1$, satisfy nonnegative linearization of products. In 2005, R. Szwarc asked to solve the analogous problem for the generalized Chebyshev polynomials $(T_n^{(α,β)}(x))_{n\in\mathbb{N}_0}$, which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure $(1-x^2)^α|x|^{2β+1}χ_{(-1,1)}(x)\,\mathrm{d}x$. In this paper, we give the solution and show that $(T_n^{(α,β)}(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products if and only if $(α,β)\in V$, so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper's original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper's one.

math.CA↗