Showing papers on "Biorthogonal system published in 1986"

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

Abstract: An asymptotic expansion including error bounds is given for polynomials {Pn, Qn} that are biorthogonal on the unit circle with respect to the weight function (1−eiθ)α+β(1−e−iθ)α−β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function2F1(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.

How good can a nonhereditary family be

Abstract: A family of vectors of a Hilbert space H is said to be hereditarily complete, if it has biorthogonal\(\mathfrak{X}'\) (minimally) and any element of H can be reconstructed from its Fourier series: . In this paper we describe all pairs of spaces A, B, which contain minimal mutually biorthogonal and complete families\(\mathfrak{X}, \mathfrak{X}' ( V\left( \mathfrak{X} \right) = A, V\left( {\mathfrak{X}'} \right) = B\) and : for this it is necessary and sufficient that the operator PAPBPA not be completely continuous. This assertion allows one to prove that: 1) if dn > 0, , then there exist an orthonormal basis {ϕn}v⩾1 and complete but not hereditarily complete biorthogonal families\(\mathfrak{X}, \mathfrak{X}'\) in H, such that ∥ Xn-ϕn∥⩽dn, ∥x′n-ϕn∥⩽ dr, (n⩾1), 2) if , then there exist families of the type described in the preceding assertion for which , where σ is any finite set of natural numbers and is the spectral projector corresponding to it. One of the auxiliary assertions is the description of all real collections α=(αk)nk=1, representable in the form , where q is a Hilbert seminorm defined in the Euclidean space En, {fk)nk=1 is a suitable orthonormal basis. This set is the convex hull of all permutations of the eigenvalues (λ1, ..., λn) of the seminorm q.