Cyclicity in Reproducing Kernel Hilbert Spaces of Analytic Functions

Abstract

Abstract. We introduce a large family of reproducing kernel Hilbert spaces H ⊂ Hol(D), which include the classical Dirichlettype spaces D α , by requiring normalized monomials to form a Riesz basis for H. Then, after precisely evaluating the n-th optimal norm and the n-th approximant of f (z) = 1−z, we completely characterize the cyclicity of functions in Hol(D) with respect to the forward shift.

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

“…If µ is a compactly supported probability measure on the real line with infinitely many points in its support, then we can form the associated sequence of monic orthogonal polynomials {P n (x)} n≥0 , where P n has degree exactly n, that is, polynomials that satisfy R P n (x)P m (x) dµ(x) = K n δ nm for some constant K n > 0. These polynomials satisfy a three term recurrence relation, which therefore gives rise to two bounded real sequences {c j } ∞ j=1 and {v j } ∞ j=1 that satisfy xP n (x) = P n+1 (x) + v n+1 P n (x) + c 2 n P n−1 (x) (13) (see [29,Theorem 1.2.3]). Favard's Theorem (see, for example, [4, p. 21]) tells us that the converse is also true, namely that given two sequences of bounded real numbers as above, with each c j > 0, there exists a compactly supported measure µ on the real line whose corresponding monic orthogonal polynomials satisfy this recursion.…”

“…It follows that the condition (12) is equivalent to the condition that 0 = λP N (λ) − ω N ω N +1 P N −1 (λ). From the recursion relation (13), this is the same as saying P N +1 (λ) = 0. We have thus determined that for j = 0, .…”

“…In [2,3], the authors were interested in norm estimates only, while here our results differ substantially for different yet equivalent norms, thus depending drastically on the geometry of the space. Cyclicity (and the concept of optimal approximant) for these spaces was studied in [13], where the role of the limit (1) is to ensure that functions analytic in a disk larger than the unit disk belong to all the spaces, and that all functions in these spaces are analytic in the unit disk. The most interesting examples of such spaces arise when ω n = (n + 1) (the Dirichlet space D), when ω n = 1 (the Hardy space H 2 ), and when ω n = (n + 1) −1 (the Bergman space A 2 ).…”

Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

“…In [3,11], it was explained that (see [11,Theorem 2.1] for the particular statement used here) the coefficients (c k ) n k=0 of the nth optimal approximant are obtained by solving the linear system M c = e 0 , (1.4) with matrix M given by (M k,l ) n k,l=0 = ( z k f, z l f α ) k,l and e 0 = ( 1, f , . .…”

A Characterization of Dirichlet-inner Functions

“…Clearly, (b) implies (a), since (a) only requires that f g Dµ ≤ g Dµ holds for a subset of all possible g ∈ D µ . Therefore what remains is to show that (a) implies D µ -inner.To see this, assume f is not D µ −inner but f Dµ = 1 and let k ∈ N\{0} such that z k f, f = 0 (12).…”