Orthogonal Polynomials of Discrete Variable and Boundedness of Dirichlet Kernel

Abstract

For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal expansions with respect to L ∞ norm, which generalize analogous results obtained for little q-Legendre, little q-Jacobi and little q-Laguerre polynomials, by the authors.

A modified Fejér and Jackson summability method with respect to orthogonal polynomials

“…If S is discrete, then there are examples with lim N →∞ ‖D N ( f ) − f ‖ ∞ = 0 for all f ∈ C(S), see [13] or [14], which is contrary to the trigonometric case. Whereas it is known since Faber [6] that in the case S = [a, b] there exist functions f ∈ C([a, b]) such that D N ( f ) does not converge towards f with respect to ‖ ‖ ∞ .…”

“…Assume that (19) does hold. Then, taking into account (14), the positivity of f implies the positivity of A N ,τ ( f ). If (19) does not hold, then there exist x 0 , y 0 ∈ S, δ < 0, and an open set U with y 0 ∈ U such that σ (N ) − n=0 τ N ,n P n (x 0 )P n (y)h n < δ for all y ∈ U.…”

The Pełczyński and Dunford–Pettis properties of the space of uniform convergent Fourier series with respect to orthogonal polynomials

“…Obviously, [4]). But there are discrete S such that U (µ) = C(S) (see [10,11,12,13]). In such a case the open mapping theorem yields the equivalence of the norms, i.e.…”