In this note, we provide simple convergence analysis for the algebraic multilevel iteration methods [37, 51]. We consider two examples of AMLI methods with different polynomial acceleration. The first one is based on shifted and scaled Chebyshev polynomial and the other on the polynomial of best approximation to x−1 on a finite interval [λmin , λmax ], 0 < λmin < λmax in the ‖ ·‖∞ norm. The con...

We evaluate explicitly the integrals ∫ 1 −1 πn(t)/(r ∓ t)dt, |r| = 1, with the πn being any one of the four Chebyshev polynomials of degree n. These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing [−1, 1] in its interior.

Three theorems are given for the integral zeros of Krawtchouk polynomials. First, five new infinite families of integral zeros for the binary (q = 2) Krawtchouk polynomials are found. Next, a lower bound is given for the next integral zero for the degree four polynomial. Finally, three new infinite families in q are found for the degree three polynomials. The techniques used are from elementary...

We give two recursive expressions for both MacWilliams and Chebyshev matrices. The expressions give rise to simple recursive algorithms for constructing the matrices. In order to derive the second recursion for the Chebyshev matrices we find out the Krawtchouk coefficients of the Discrete Chebyshev polynomials, a task interesting on its own.

i=0 (n − i)‖p‖∞, p(x) ∈ Πn, 1 ≤ k ≤ n are typical examples of inequalities connecting norms of a polynomial and its derivatives. In (1.1) the equality holds only at x = ±1 and only when p(x) = cTn(x), where Tn(x) is the Chebyshev polynomial of the first kind of a degree n c © 2011 Mathematical Institute, Slovak Academy of Sciences. 2010 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: 33C...

We consider the condition of orthogonal polynomials, encoded by the coeecients of their three-term recurrence relation, if the measure is given by modiied moments (i.e. integrals of certain polynomials forming a basis). The results concerning various polynomial bases are illustrated with simple examples of generating (possibly shifted) Chebyshev polynomials of rst and second kind.

The family of general Jacobi polynomials P (α,β) n where α, β ∈ C can be characterised by complex (nonhermitian) orthogonality relations (cf. [15]). The special subclass of Jacobi polynomials P (α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another spe...

Starting from a sequence {pn{x; no)} of orthogonal polynomials with an orthogonality measure yurj supported on Eo C [—1,1], we construct a new sequence {p„(x;fi)} of orthogonal polynomials on£ = T~1(Eq) (T is a polynomial of degree TV) with an orthogonality measure [i that is related to noIf Eo = [—1,1], then E = T_1([-l,l]) will in general consist of TV intervals. We give explicit formulas rel...

Littlewood asked how small the ratio ||f || 4 /||f || 2 (where ||·|| α denotes the L α norm on the unit circle) can be for polynomials f having all coefficients in {1, −1}, as the degree tends to infinity. Since 1988, the least known asymptotic value of this ratio has been 4 7/6, which was conjectured to be minimum. We disprove this conjecture by showing that there is a sequence of such polynom...

In (West, Discrete Math. 157 (1996) 363-374) it was shown using transfer matrices that the number [Sn(123; 3214)1 of permutations avoiding the pattems 123 and 3214 is the Fibonacci number F2, (as are also IS,(213; 1234)1 and 1S~(213;4123)1 ). We now find the transfer matrix for IS , (123;r , r 1 . . . . . 2,1,r + 1)1, IS,(213;1,2 . . . . . r , r + 1)1, and ISn(213;r + 1,1,2 . . . . . r)l, deter...