Showing papers on "Biorthogonal system published in 1987"

Nonharmonic Fourier series and spectral theory

Abstract: We consider the problem of using functions g" (x) exp (iX,1 x) to form biorthogonal expansions in the spaces LP(-7r, sr), for various values of p. The work of Paley and Wiener and of Levinson considered conditions of the form IX,, nI < A (p) which insure that {g,,} is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for p = 2. In this paper, rather than imposing an explicit growth condition, we assume that { X, n } is a multiplier sequence on LP (-T, 7T). Conditions are given insuring that { g,, } inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, Xn and gn are shown to be the eigenvalues and eigenfunctions of an unbounded operator A which is closely related to a differential operator, i A generates a strongly continuous group and -A2 generates a strongly continuous semigroup. Half-range expansions, involving cos X, x or sin X n x on (0, 7T) are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.

On a biorthogonal system associated with uniform asymptotic expansions

Abstract: In the present paper, a family of rational functions is introduced, and the two families form a biorthogonal system on a contour in the complex plane. The system can be viewed as a generalization of the families {z n } and {z −n−1 }, which occur in Taylor expansions and the Cauchy integrals of analytic functions. Explicit representations of the rational functions are given together with the rigorous estimates

On the q-Konhauser biorthogonal polynomials

Abstract: Recently Al-Salam and Verma discussed two polynomial sets {Zn(α)(x,k|q)} and {Yn(α)(x,k|q)} which are biorthogonal on (0,∞) with respect to a continuous or discrete distribution function. For the polynomials Yn(α)(x,k|q) the operational formula is derived.