A continuous function space with a Faber basis

Abstract

Let SCR be compact with #S ¼ N and let CðSÞ be the set of all real continuous functions on S: We ask for an algebraic polynomial sequence ðP n Þ N n¼0 with deg P n ¼ n such that every f ACðSÞ has a unique representation f ¼ P N i¼0 a i P i and call such a basis Faber basis. In the special case of S ¼ S q ¼ fq k ; kAN 0 g,f0g; 0oqo1; we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with AðS q ÞaUðS q Þ ¼ CðS q Þ; where AðS q Þ is the so-called Wiener algebra and UðS q Þ is the set of all f ACðS q Þ which are uniquely represented by its Fourier series. r

Orthogonal Polynomials of Discrete Variable and Boundedness of Dirichlet Kernel

“…Theorem 1. Assume (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) are satisfied. Then the orthogonality measure µ is concentrated on decreasing sequence {ξ n } ∞ n=1 , where ξ n ∼ q n , the quantity 1 − ξ n+1 /ξ n is bounded away from 0, and µ([0, ξ n ]) ∼ s −n .…”

“…Little q-Legendre polynomials, more generally q-Jacobi polynomials and little q-Laguerre polynomials are such. The uniform boundedness of s n (f ) ∞ have been shown for these systems in [6,7,8]. The proof depended heavily on the precise knowledge of the orthogonality measure and pointwise estimates of these polynomials.…”

“…(6) we gets n (f, x) = S f (y) n k=0 R k (x)R k (y)h(k) dµ(y) = S f (y)n k=0 p k (x)p k (y) dµ(y). Define the generalized Dirichlet kernel K n (x, y) by (29) K n (x, y) = n k=0 p k (x)p k (y).…”

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 ‖ ‖ ∞ .…”

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.…”