Let μ be a positive Borel measure on R such that the polynomials belong to L_p (R,dμ). A necessar... more Let μ be a positive Borel measure on R such that the polynomials belong to L_p (R,dμ). A necessary and sufficient condition is given for the polynomials to be dense in L_p (R,dμ). It reads as follows: the measure μ should be representable as dμ=w^p dv, with a finite Borel measure v and an upper semicontinuous function w:R→[0,1] such that the polynomials belong to and are dense in the space C_0^w={f∈C(R): f(x)w(x)→0 as |x|→∞}. The proof uses the Mergelyan's criterion for the density of polynomials in L_p spaces.
The known theorem of A. Borichev and M. Sodin [J. Anal. Math. 76, 219–264] on the description of ... more The known theorem of A. Borichev and M. Sodin [J. Anal. Math. 76, 219–264] on the description of sets of elements of the second line of the Nevanlinna matrices which correspond to indefinite moment problems is extended to the sets of elements of the first line of these matrices.
This paper is devoted to application of the results about entire and meromorphic functions obtain... more This paper is devoted to application of the results about entire and meromorphic functions obtained by A.Baernstein II (1974), I.V. Ostrowski (1970) and W.K. Hayman (1989) for weakening of the validity conditions for partial fraction decomposition of the reciprocal of an entire function which have earlier been found by M.G. Krein (1947), B.Ja.Levin (1956), L.de Branges (1959), P.Koosis (1988) and H.Pedersen (1995).
The criterion for the denseness of polynomials in the space L_2 ( R , d μ ) established by Hambur... more The criterion for the denseness of polynomials in the space L_2 ( R , d μ ) established by Hamburger in 1921 is extended to the spaces L_p ( R , d μ ) , 1 ≤ p < ∞ .
A calculation formula is established for the codimension of the polynomial subspace in L_2 (R; dμ... more A calculation formula is established for the codimension of the polynomial subspace in L_2 (R; dμ) with discrete indeterminate measure μ. We clarify how much the masspoint of the n-canonical solution of an indeterminate Hamburger moment problem differs from the masspoint of the corresponding N-extremal solution at a given point of the real axis.
We prove that the theorem on the incompleteness of polynomials in the space C w 0 established by ... more We prove that the theorem on the incompleteness of polynomials in the space C w 0 established by de Branges in 1959 is not true for the space L d p ( , ) R μ if the support of the measure μ is sufficiently dense.
One of the basic formulas of subdifferential calculus is the Moreau--Rockafellar formula 0 (p + q... more One of the basic formulas of subdifferential calculus is the Moreau--Rockafellar formula 0 (p + q) = Op + ~q
Proceedings of the American Mathematical Society, 2008
It has been proved that algebraic polynomials P are dense in the space L p (R, dµ), p ∈ (0, ∞), i... more It has been proved that algebraic polynomials P are dense in the space L p (R, dµ), p ∈ (0, ∞), iff the measure µ is representable as dµ = w p dν with a finite non-negative Borel measure ν and an upper semi-continuous function w :
The following problem, suggested by Laguerre's Theorem (1884), remains open: Characterize all... more The following problem, suggested by Laguerre's Theorem (1884), remains open: Characterize all real sequences fkg1 k=0 which have the zero-di- minishing property; that is, if p(x )= P n k=0akx k is any real polynomial, then Pn k=0kakx k has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak
A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribut... more A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribution of real zeros of a polynomial is to characterize all real sequences T={γ_k}_{k=0}^∞ such that they satisfy the property Z_c (T[p(x)])≤Z_c (p(x)), where Z_c(p(x)) denotes the number of non-real zeros of the real polynomial p(x)=∑_{k=0}^{n} a_k x_k and T(p(x))=∑_{k=0}^{n}γ_k a_k x_k. The main result of this paper shows that under a mild growth restriction, an entire function of exponential type f(z) for which the sequence T={f(k)}_{k=0}^{∞} satisfies the above condition must have only real zeros. The paper concludes with some applications to the Riemann hypothesis.
Proceedings of the American Mathematical Society, 2008
We describe all zero-diminishing sequences (over the real-valued polynomials on R) which addition... more We describe all zero-diminishing sequences (over the real-valued polynomials on R) which additionally satisfy a Carleman condition and show that they are of the same kind as those in E. Laguerre's theorem from 1884.
The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all rea... more The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all real sequences {μ k } k=0 ∞ which have the zero-diminishing property; that is, if p(x)=∑ k=0 n a k x k is any real polynomial, then ∑ k=0 n μ k a k x k has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak growth condition on the sequence {μ k } k=0 ∞ , namely lim ̲ n→∞ |μ n | 1/n <∞. More precisely, it is established that the real sequence {μ k } k≥0 is a weakly increasing zero-diminishing sequence if and only if there exists σ∈{+1,-1} and an entire function Φ(z)=be az ∏ n≥1 1+x α n ,a,b∈R 1 ,b≠0,α n >0,∀n≥1,∑ n≥1 1 α n <∞, such that μ k =σ k /Φ(k),∀k≥0.
Proceedings of the American Mathematical Society, 2014
A deep result of J. shows that the polylogarithms Li α (z) := ∞ k=1 z k /k α map the open unit di... more A deep result of J. shows that the polylogarithms Li α (z) := ∞ k=1 z k /k α map the open unit disk D centered at the origin oneto-one onto convex domains for all α ≥ 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain Λ := C \ [1, ∞), introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that Li α (z) maps an arbitrary open disk or half-plane in Λ one-to-one onto a convex domain for every α ≥ 1.
We extend the property (N) introduced by Jameson for closed convex cones to the normal property f... more We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex
Let μ be a positive Borel measure on R such that the polynomials belong to L_p (R,dμ). A necessar... more Let μ be a positive Borel measure on R such that the polynomials belong to L_p (R,dμ). A necessary and sufficient condition is given for the polynomials to be dense in L_p (R,dμ). It reads as follows: the measure μ should be representable as dμ=w^p dv, with a finite Borel measure v and an upper semicontinuous function w:R→[0,1] such that the polynomials belong to and are dense in the space C_0^w={f∈C(R): f(x)w(x)→0 as |x|→∞}. The proof uses the Mergelyan's criterion for the density of polynomials in L_p spaces.
The known theorem of A. Borichev and M. Sodin [J. Anal. Math. 76, 219–264] on the description of ... more The known theorem of A. Borichev and M. Sodin [J. Anal. Math. 76, 219–264] on the description of sets of elements of the second line of the Nevanlinna matrices which correspond to indefinite moment problems is extended to the sets of elements of the first line of these matrices.
This paper is devoted to application of the results about entire and meromorphic functions obtain... more This paper is devoted to application of the results about entire and meromorphic functions obtained by A.Baernstein II (1974), I.V. Ostrowski (1970) and W.K. Hayman (1989) for weakening of the validity conditions for partial fraction decomposition of the reciprocal of an entire function which have earlier been found by M.G. Krein (1947), B.Ja.Levin (1956), L.de Branges (1959), P.Koosis (1988) and H.Pedersen (1995).
The criterion for the denseness of polynomials in the space L_2 ( R , d μ ) established by Hambur... more The criterion for the denseness of polynomials in the space L_2 ( R , d μ ) established by Hamburger in 1921 is extended to the spaces L_p ( R , d μ ) , 1 ≤ p < ∞ .
A calculation formula is established for the codimension of the polynomial subspace in L_2 (R; dμ... more A calculation formula is established for the codimension of the polynomial subspace in L_2 (R; dμ) with discrete indeterminate measure μ. We clarify how much the masspoint of the n-canonical solution of an indeterminate Hamburger moment problem differs from the masspoint of the corresponding N-extremal solution at a given point of the real axis.
We prove that the theorem on the incompleteness of polynomials in the space C w 0 established by ... more We prove that the theorem on the incompleteness of polynomials in the space C w 0 established by de Branges in 1959 is not true for the space L d p ( , ) R μ if the support of the measure μ is sufficiently dense.
One of the basic formulas of subdifferential calculus is the Moreau--Rockafellar formula 0 (p + q... more One of the basic formulas of subdifferential calculus is the Moreau--Rockafellar formula 0 (p + q) = Op + ~q
Proceedings of the American Mathematical Society, 2008
It has been proved that algebraic polynomials P are dense in the space L p (R, dµ), p ∈ (0, ∞), i... more It has been proved that algebraic polynomials P are dense in the space L p (R, dµ), p ∈ (0, ∞), iff the measure µ is representable as dµ = w p dν with a finite non-negative Borel measure ν and an upper semi-continuous function w :
The following problem, suggested by Laguerre's Theorem (1884), remains open: Characterize all... more The following problem, suggested by Laguerre's Theorem (1884), remains open: Characterize all real sequences fkg1 k=0 which have the zero-di- minishing property; that is, if p(x )= P n k=0akx k is any real polynomial, then Pn k=0kakx k has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak
A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribut... more A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribution of real zeros of a polynomial is to characterize all real sequences T={γ_k}_{k=0}^∞ such that they satisfy the property Z_c (T[p(x)])≤Z_c (p(x)), where Z_c(p(x)) denotes the number of non-real zeros of the real polynomial p(x)=∑_{k=0}^{n} a_k x_k and T(p(x))=∑_{k=0}^{n}γ_k a_k x_k. The main result of this paper shows that under a mild growth restriction, an entire function of exponential type f(z) for which the sequence T={f(k)}_{k=0}^{∞} satisfies the above condition must have only real zeros. The paper concludes with some applications to the Riemann hypothesis.
Proceedings of the American Mathematical Society, 2008
We describe all zero-diminishing sequences (over the real-valued polynomials on R) which addition... more We describe all zero-diminishing sequences (over the real-valued polynomials on R) which additionally satisfy a Carleman condition and show that they are of the same kind as those in E. Laguerre's theorem from 1884.
The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all rea... more The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all real sequences {μ k } k=0 ∞ which have the zero-diminishing property; that is, if p(x)=∑ k=0 n a k x k is any real polynomial, then ∑ k=0 n μ k a k x k has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak growth condition on the sequence {μ k } k=0 ∞ , namely lim ̲ n→∞ |μ n | 1/n <∞. More precisely, it is established that the real sequence {μ k } k≥0 is a weakly increasing zero-diminishing sequence if and only if there exists σ∈{+1,-1} and an entire function Φ(z)=be az ∏ n≥1 1+x α n ,a,b∈R 1 ,b≠0,α n >0,∀n≥1,∑ n≥1 1 α n <∞, such that μ k =σ k /Φ(k),∀k≥0.
Proceedings of the American Mathematical Society, 2014
A deep result of J. shows that the polylogarithms Li α (z) := ∞ k=1 z k /k α map the open unit di... more A deep result of J. shows that the polylogarithms Li α (z) := ∞ k=1 z k /k α map the open unit disk D centered at the origin oneto-one onto convex domains for all α ≥ 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain Λ := C \ [1, ∞), introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that Li α (z) maps an arbitrary open disk or half-plane in Λ one-to-one onto a convex domain for every α ≥ 1.
We extend the property (N) introduced by Jameson for closed convex cones to the normal property f... more We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex
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