We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)... more We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)f (z + h) + M 2 (z)f (z − h) preserving the Laguerre-Pólya class of entire functions. Here M 1 and M 2 are some complex functions and h is a nonzero complex number.
Computational Methods and Function Theory, Sep 29, 2006
In the paper it is discussed the relations of the Riemann ζ−function to classes of generating fun... more In the paper it is discussed the relations of the Riemann ζ−function to classes of generating functions of multiply positive sequences according to Schoenberg (also called Pólya frequency sequences).
We show that the constant $\frac{1}{cos^2(\frac{\pi}{n+2})}$ in this theorem could not be increas... more We show that the constant $\frac{1}{cos^2(\frac{\pi}{n+2})}$ in this theorem could not be increased. We also present some corollaries of this theorem.
This note deals with linear operators preserving the set of positive (nonnegative) polynomials. N... more This note deals with linear operators preserving the set of positive (nonnegative) polynomials. Numerous works of prominent mathematicians in fact contain the exhaustive description of linear operators preserving the set of positive (nonnegative) polynomials. In spite of this, since this description was not formulated explicitly, it is almost lost for possible applications. In the paper we formulate and prove these classical results and give some applications. For example, we prove that there are no linear ordinary differential operators of order m ∈ AE with polynomial coefficients which map the set of nonnegative (positive) polynomials of degree ≤ (m 2 + 1) into the set of nonnegative polynomials. This result is a generalization of a Theorem by Guterman and Shapiro.
Complex Variables and Elliptic Equations, Nov 16, 2017
We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)... more We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)f (z + h) + M 2 (z)f (z − h) preserving the Laguerre-Pólya class of entire functions. Here M 1 and M 2 are some complex functions and h is a nonzero complex number.
The following theorem is proved. Theorem. Suppose M = (a i,j) be a k × k matrix with positive ent... more The following theorem is proved. Theorem. Suppose M = (a i,j) be a k × k matrix with positive entries and a i,j a i+1,j+1 > 4 cos 2 π k+1 a i,j+1 a i+1,j (1 ≤ i ≤ k − 1, 1 ≤ j ≤ k − 1). Then det M > 0. The constant 4 cos 2 π k+1 in this Theorem is sharp. A few other results concerning totally positive and multiply positive matrices are obtained.
ABSTRACT Let x 1,..., x n and y 1,..., y n be nonnegative numbers. If the l p -norm of (x 1,..., ... more ABSTRACT Let x 1,..., x n and y 1,..., y n be nonnegative numbers. If the l p -norm of (x 1,..., x n ) coincides with that of (y 1,..., y n ) for n different values of p > 0, then the x i are a permutation of the x i. This generalizes a well-known algebraic result which concerns the exponents p = 1,..., n.¶Some consequences as well as some generalizations to infinite sequences and to functions are also discussed.
Journal of Mathematical Analysis and Applications, Nov 1, 2008
A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a giv... more A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n ∈ N we find the smallest possible constant d n > 0 such that if the coefficients of F (z) = a 0 + a 1 z + • • • + a n z n are positive and satisfy the inequalities a k a k+1 > d n a k−1 a k+2 for k = 1, 2,. .. ,n − 2, then F (z) is Hurwitz.
We study the effect of finite difference operators of finite order on the distribution of zeros o... more We study the effect of finite difference operators of finite order on the distribution of zeros of polynomials and entire functions.
Theorem 1 A set E = {b k } ∞ k=1 ⊂ C without finite accumulation points is the zero set of an ent... more Theorem 1 A set E = {b k } ∞ k=1 ⊂ C without finite accumulation points is the zero set of an entire function bounded in C ω , ∀ω ∈ R, iff b k ∈E C ω |Reb k | + 1 |b k | 2 + 1 < ∞, ∀ω ∈ R. (0.8) Note that the necessity of the condition (0.8) is an easy consequence of the well-known Blaschke condition for a half-plane. It can be easily shown that the condition (0.8) implies (b) in Theorem A. It turned out that, if we add (0.8) to the conditions of Theorem A, we do not obtain the complete charactreization of zero sets of entire absolutely monotonic functions. In [1] it was mentioned the following necessary condition not depending on all previous ones: dist(x, E) → +∞, x → −∞. (0.9) I.V. Ostrovskii showed (oral communication) that the following independent condition is also necessary:
A new necessary condition for a set to be the zero set of an absolutely monotonic function is giv... more A new necessary condition for a set to be the zero set of an absolutely monotonic function is given. If A ⊂ {z : Im z < 0} is the zero set of an absolutely monotonic function then for any β ∈ (0, π/2) there exists a nonnegative continuous function h β , h β ∈ L 1 (−∞, −1], such that a∈A {a:| arg a−π|<β}
We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with... more We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form ∆ θ,h (f)(z) = e iθ f (z + ih) − e −iθ f (z − ih), θ ∈ [0, π), h ∈ C \ {0}, where f is a polynomial or an entire function of a certain kind, and prove that the roots of ∆ θ,h (f) are simple under some conditions. Moreover, we prove that the operator ∆ θ,h does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of ∆ θ,h (p) as |h| → ∞ is found for any complex polynomial p. Some other interesting roots preserving properties of the operator ∆ θ,h are also studied, and a few examples are presented.
We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A showing that ... more We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A showing that inf max ± 1 ± x − 1 : x ∈ A, x = ε π 4e ε 2 + o(ε 2) as ε → 0. We also give a characterization of two-dimensional subspaces of Banach algebras containing the identity in terms of polynomial inequalities.
In this Note we prove a new result about (finite) multiplier sequences, i.e. linear operators act... more In this Note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of R[x] and sending polynomials with all real roots to polynomials with all real roots. Namely, we show that any such operator does not decrease the logarithmic mesh when acting on an arbitrary polynomial having all roots real and of the same sign. The logarithmic mesh of such a polynomial is defined as the minimal quotient of its consecutive roots taken in the non-decreasing order of their absolute values.
The problem of whether an entire function f with nonnegative coefficients can be multiplied by an... more The problem of whether an entire function f with nonnegative coefficients can be multiplied by another entire function g which “grows not faster” than f so that the product fg is a generating function of a Pólya frequency sequence of finite order (multipositive sequence in the sense of Fekete) is investigated.
We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with... more We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form ∆ θ,h (f)(z) = e iθ f (z + ih) − e −iθ f (z − ih), θ ∈ [0, π), h ∈ C \ {0}, where f is a polynomial or an entire function of a certain kind, and prove that the roots of ∆ θ,h (f) are simple under some conditions. Moreover, we prove that the operator ∆ θ,h does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of ∆ θ,h (p) as |h| → ∞ is found for any complex polynomial p. Some other interesting roots preserving properties of the operator ∆ θ,h are also studied, and a few examples are presented.
A new necessary condition for a set to be the zero set of an absolutely monotonic function is giv... more A new necessary condition for a set to be the zero set of an absolutely monotonic function is given. If A⊂{z:ℜz<0} is the zero set of an absolutely monotonic function, then for any β∈(0,π/2) there exists a nonnegative continuous function h β , h β ∈L 1 (-∞,-1], such that ∑ a∈A⋂{a:|arga-π|<β} 1 (x-a) 2 ≤h β (x)· It is shown that this condition is not a consequence of conditions known before.
We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)... more We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)f (z + h) + M 2 (z)f (z − h) preserving the Laguerre-Pólya class of entire functions. Here M 1 and M 2 are some complex functions and h is a nonzero complex number.
Computational Methods and Function Theory, Sep 29, 2006
In the paper it is discussed the relations of the Riemann ζ−function to classes of generating fun... more In the paper it is discussed the relations of the Riemann ζ−function to classes of generating functions of multiply positive sequences according to Schoenberg (also called Pólya frequency sequences).
We show that the constant $\frac{1}{cos^2(\frac{\pi}{n+2})}$ in this theorem could not be increas... more We show that the constant $\frac{1}{cos^2(\frac{\pi}{n+2})}$ in this theorem could not be increased. We also present some corollaries of this theorem.
This note deals with linear operators preserving the set of positive (nonnegative) polynomials. N... more This note deals with linear operators preserving the set of positive (nonnegative) polynomials. Numerous works of prominent mathematicians in fact contain the exhaustive description of linear operators preserving the set of positive (nonnegative) polynomials. In spite of this, since this description was not formulated explicitly, it is almost lost for possible applications. In the paper we formulate and prove these classical results and give some applications. For example, we prove that there are no linear ordinary differential operators of order m ∈ AE with polynomial coefficients which map the set of nonnegative (positive) polynomials of degree ≤ (m 2 + 1) into the set of nonnegative polynomials. This result is a generalization of a Theorem by Guterman and Shapiro.
Complex Variables and Elliptic Equations, Nov 16, 2017
We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)... more We completely describe all finite difference operators of the form ∆ M 1 ,M 2 ,h (f)(z) = M 1 (z)f (z + h) + M 2 (z)f (z − h) preserving the Laguerre-Pólya class of entire functions. Here M 1 and M 2 are some complex functions and h is a nonzero complex number.
The following theorem is proved. Theorem. Suppose M = (a i,j) be a k × k matrix with positive ent... more The following theorem is proved. Theorem. Suppose M = (a i,j) be a k × k matrix with positive entries and a i,j a i+1,j+1 > 4 cos 2 π k+1 a i,j+1 a i+1,j (1 ≤ i ≤ k − 1, 1 ≤ j ≤ k − 1). Then det M > 0. The constant 4 cos 2 π k+1 in this Theorem is sharp. A few other results concerning totally positive and multiply positive matrices are obtained.
ABSTRACT Let x 1,..., x n and y 1,..., y n be nonnegative numbers. If the l p -norm of (x 1,..., ... more ABSTRACT Let x 1,..., x n and y 1,..., y n be nonnegative numbers. If the l p -norm of (x 1,..., x n ) coincides with that of (y 1,..., y n ) for n different values of p &gt; 0, then the x i are a permutation of the x i. This generalizes a well-known algebraic result which concerns the exponents p = 1,..., n.¶Some consequences as well as some generalizations to infinite sequences and to functions are also discussed.
Journal of Mathematical Analysis and Applications, Nov 1, 2008
A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a giv... more A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n ∈ N we find the smallest possible constant d n > 0 such that if the coefficients of F (z) = a 0 + a 1 z + • • • + a n z n are positive and satisfy the inequalities a k a k+1 > d n a k−1 a k+2 for k = 1, 2,. .. ,n − 2, then F (z) is Hurwitz.
We study the effect of finite difference operators of finite order on the distribution of zeros o... more We study the effect of finite difference operators of finite order on the distribution of zeros of polynomials and entire functions.
Theorem 1 A set E = {b k } ∞ k=1 ⊂ C without finite accumulation points is the zero set of an ent... more Theorem 1 A set E = {b k } ∞ k=1 ⊂ C without finite accumulation points is the zero set of an entire function bounded in C ω , ∀ω ∈ R, iff b k ∈E C ω |Reb k | + 1 |b k | 2 + 1 < ∞, ∀ω ∈ R. (0.8) Note that the necessity of the condition (0.8) is an easy consequence of the well-known Blaschke condition for a half-plane. It can be easily shown that the condition (0.8) implies (b) in Theorem A. It turned out that, if we add (0.8) to the conditions of Theorem A, we do not obtain the complete charactreization of zero sets of entire absolutely monotonic functions. In [1] it was mentioned the following necessary condition not depending on all previous ones: dist(x, E) → +∞, x → −∞. (0.9) I.V. Ostrovskii showed (oral communication) that the following independent condition is also necessary:
A new necessary condition for a set to be the zero set of an absolutely monotonic function is giv... more A new necessary condition for a set to be the zero set of an absolutely monotonic function is given. If A ⊂ {z : Im z < 0} is the zero set of an absolutely monotonic function then for any β ∈ (0, π/2) there exists a nonnegative continuous function h β , h β ∈ L 1 (−∞, −1], such that a∈A {a:| arg a−π|<β}
We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with... more We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form ∆ θ,h (f)(z) = e iθ f (z + ih) − e −iθ f (z − ih), θ ∈ [0, π), h ∈ C \ {0}, where f is a polynomial or an entire function of a certain kind, and prove that the roots of ∆ θ,h (f) are simple under some conditions. Moreover, we prove that the operator ∆ θ,h does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of ∆ θ,h (p) as |h| → ∞ is found for any complex polynomial p. Some other interesting roots preserving properties of the operator ∆ θ,h are also studied, and a few examples are presented.
We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A showing that ... more We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A showing that inf max ± 1 ± x − 1 : x ∈ A, x = ε π 4e ε 2 + o(ε 2) as ε → 0. We also give a characterization of two-dimensional subspaces of Banach algebras containing the identity in terms of polynomial inequalities.
In this Note we prove a new result about (finite) multiplier sequences, i.e. linear operators act... more In this Note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of R[x] and sending polynomials with all real roots to polynomials with all real roots. Namely, we show that any such operator does not decrease the logarithmic mesh when acting on an arbitrary polynomial having all roots real and of the same sign. The logarithmic mesh of such a polynomial is defined as the minimal quotient of its consecutive roots taken in the non-decreasing order of their absolute values.
The problem of whether an entire function f with nonnegative coefficients can be multiplied by an... more The problem of whether an entire function f with nonnegative coefficients can be multiplied by another entire function g which “grows not faster” than f so that the product fg is a generating function of a Pólya frequency sequence of finite order (multipositive sequence in the sense of Fekete) is investigated.
We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with... more We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form ∆ θ,h (f)(z) = e iθ f (z + ih) − e −iθ f (z − ih), θ ∈ [0, π), h ∈ C \ {0}, where f is a polynomial or an entire function of a certain kind, and prove that the roots of ∆ θ,h (f) are simple under some conditions. Moreover, we prove that the operator ∆ θ,h does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of ∆ θ,h (p) as |h| → ∞ is found for any complex polynomial p. Some other interesting roots preserving properties of the operator ∆ θ,h are also studied, and a few examples are presented.
A new necessary condition for a set to be the zero set of an absolutely monotonic function is giv... more A new necessary condition for a set to be the zero set of an absolutely monotonic function is given. If A⊂{z:ℜz<0} is the zero set of an absolutely monotonic function, then for any β∈(0,π/2) there exists a nonnegative continuous function h β , h β ∈L 1 (-∞,-1], such that ∑ a∈A⋂{a:|arga-π|<β} 1 (x-a) 2 ≤h β (x)· It is shown that this condition is not a consequence of conditions known before.
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