We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$ , where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds, in terms of the minimal polynomials of $\cos ({2\pi }{/d})$ . The method of proof is based on earlier work, D. A. Wolfram, [‘Factoring variants of Chebyshev polynomials of the first and second kinds with minimal polynomials of $\cos ({2 \pi }/{d})$ ’, Amer. Math. Monthly 129 (2022), 172–176] for factoring variants of Chebyshev polynomials of the first and second kinds. We extend this to show that, in general, similar variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x) \pm 1$ and $Y_n(x) \pm 1$ , do not have factors that are minimal polynomials of $\cos ({2\pi }/{d})$ .

We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.

For symmetric eigenvalue problems, we constructed a three-term recurrence polynomial filter by means of Chebyshev polynomials. The new filtering technique does not need to solve linear systems and only needs matrix-vector products. It is a memory conserving filtering technique for its three-term recurrence relation. As an application, we use this filtering strategy to the Davidson method and propose the filtered-Davidson method. Through choosing suitable shifts, this method can gain cubic convergence rate locally. Theory and numerical experiments show the efficiency of the new filtering technique.

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

The maximum entropy method for the Hausdorff moment problem suffers from ill conditioning as it uses monomial basis {1,x,x2,...,xn}. The maximum entropy method for the Chebyshev moment probelm was studied to overcome this drawback in. In this paper we review and modify the maximum entropy method for the Hausdorff and Chebyshev moment problems studied in and present the maximum entropy method for the Legendre moment problem. We also give the algorithms of converting the Hausdorff moments into the Chebyshev and Lengendre moments, respectively, and utilizing the corresponding maximum entropy method.

We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes, and V. Trevisan.

Let S: [0, 1]→[0, 1] be a chaotic map and let f* be a stationary density of the Frobenius-Perron operator PS: L1→L1 associated with S. We develop a numerical algorithm for approximating f*, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.

It is shown that the dnth Chebyshev polynomials on the Julia set JP, and on the equipotential curve ΓP(R), of the polynomial P(z)=zd−c, are identical and exactly equal to the nth iteration of P(z) itself. As an application, the capacity of the Julia set JP is deduced, a result that was first obtained by Brolin.

This paper considers the rational system ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})\,\,:=\,\left\{ \frac{P(x)}{\Pi _{k=1}^{n}(x-{{a}_{k}})},\,P\,\in \,{{P}_{n}} \right\}$ with nonreal elements in $\left\{ {{a}_{k}} \right\}_{k=1}^{n}\,\subset \,\mathbb{C}\,\backslash \,[-1,\,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results

Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,\,p,\,n)$ were constructed recently in the work of Ariki and Koike [AK], Broué and Malle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,\,p,\,n)$ given by Ariki and Koike [AK] and Ariki [Ari].

This paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine

Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.

Let be the class of all polynomials/? of degree at most n such that In view of the example zn it follows from Bernstein's inequality for polynomials that at each point z0 of the unitcirle. It was shown by A. Giroux and Q. I. Rahman [2] that if denotes the subclass of polynomials in which vanish at 1, then

where c1 and c2 are constants not depending on n. Here we find the exact value of which has some special significance and also at certain other points of the unit circle.

We consider some variations of Muntz's classical theorem on when Span{xλi} is dense in C[0,1]. We prove, for example, that if then the collection of spaces is dense in C[0,1] if and only if lim sup(log n)/λn = ∞. Another variation concerns the denseness of the union of spaces of the form The derivations of these results require an examination of the location of the zeros of the associated Chebyshev polynomials.

Each polynomial P(x) has a "Lorentz representation", of the form This representation becomes unique if we insist that n equals the degree of P. Motivated partly by questions involving polynomials with integer coefficients, we investigate the relationship between