Suppose that {λ n } ∞ n=1 is a sequence of distinct positive real numbers satisfying the conditio... more Suppose that {λ n } ∞ n=1 is a sequence of distinct positive real numbers satisfying the conditions inf{λ n+1 − λ n } > 0, and ∞ n=1 λ −1 n < ∞. We prove that the exponential system {e λnt } ∞ n=1 is hereditarily complete in the closure of the subspace spanned by {e λnt } ∞ n=1 in the space L 2 (a, b). We also give an example of a class of compact non-normal operators defined on this closure which admit spectral synthesis.
We prove that if t u 8 "1 is a complete Riesz-Fischer sequence in a separable Hilbert space , the... more We prove that if t u 8 "1 is a complete Riesz-Fischer sequence in a separable Hilbert space , then :" t P : ÿ |x , y| 2 ă 8u is closed in if and only if t u 8 "1 has a biorthogonal Riesz sequence. If the latter is also complete in , then t u 8 "1 is a Riesz basis for .
Let {e iλnt } n∈Z be an exponential Schauder Basis for L 2 (0, 1), for λ n ∈ R, and let {r n (t)}... more Let {e iλnt } n∈Z be an exponential Schauder Basis for L 2 (0, 1), for λ n ∈ R, and let {r n (t)} n∈Z be its dual Schauder Basis. Let A be a non-empty subset of the integers containing exactly M elements. We prove that for α > 0 the weighted system {t α • r n (t)} n∈Z\A is exact in the space L 2 (0, 1), that is, it is complete and minimal in L 2 (0, 1), if and only if M − 1 2 ≤ α < M + 1 2. We also show that such a system is not a Riesz Basis for L 2 (0, 1). In particular, the weighted trigonometric system {t α • e 2πint } n∈Z\A is exact in L 2 (0, 1), if and only if α ∈ [M − 1 2 , M + 1 2), but it is not a Schauder Basis for L 2 (0, 1).
In this paper we prove that if U = {u n } is a minimal sequence in a separable Hilbert space H , ... more In this paper we prove that if U = {u n } is a minimal sequence in a separable Hilbert space H , then multiplying each vector u n by an appropriate constant c n , yields a family of functions P := {c n • u n } , such that P is a Riesz-Fischer sequence in H. If U is a minimal and complete sequence in H , therefore having a unique biorthogonal sequence V = {v n } in H , then P is a complete Riesz-Fischer sequence in H and its unique biorthogonal family {v n /c n } is a Bessel sequence in H .
Let {λn} ∞ n=1 be a strictly increasing sequence of positive real numbers diverging to infinity a... more Let {λn} ∞ n=1 be a strictly increasing sequence of positive real numbers diverging to infinity and let {μn} ∞ n=1 be a sequence of positive integers. Consider the exponential system EΛ = {t k e λnt : k = 0, 1, 2,. .. , μn − 1} ∞ n=1. Assuming the density condition lim t→∞ λn≤t μn t = d < ∞ and some other restrictions, we prove that every function in the closure of the linear span of EΛ in some weighted Banach spaces on the real line R is extended to an entire function represented by a Taylor-Dirichlet series g(z) = ∞ n=1 μn −1 k=0 c n,k z k e λnz , c n,k ∈ C. We also consider a problem in a weighted L 2 (R) Hilbert space as well as a moment problem on the real line.
The closed span of some Exponential system E Λ in the spaces L p (γ, β), properties of a Biorthog... more The closed span of some Exponential system E Λ in the spaces L p (γ, β), properties of a Biorthogonal family to E Λ in L 2 (γ, β), Moment problems, and a differential equation of Carleson
In this article we give new criteria for two complex sequences to have the same excess in the sen... more In this article we give new criteria for two complex sequences to have the same excess in the sense of Paley and Wiener in L 2 (−a, a). As a result, we prove that given any positive integer q, a real
Perturbation of zeros of entire functions of exponential type
This thesis is divided into two parts. In the first part, we give new criteria for two complex se... more This thesis is divided into two parts. In the first part, we give new criteria for two complex sequences to have the same excess in the sense of Paley and Wiener in L2(−a, a). As a result, we prove that given any positive integer q, a real number α ∈ ( 0, 1 2π ) and complex numbers ν0 = 0, νn = nq + iα log |nq|, |n| ≥ 1, the exponential system {tkeitνn : k = 0, 1, . . . , q − 1}n=−∞ has excess 0 in L2(−π, π). In the second part of the thesis, we give an extension of a theorem of N. Levinson (see Theorems 3.1 and 3.2). As an application, we get a variation of the Fabry Gap Theorem for frequencies with finite upper density (see Theorems 3.3 and 3.4), concerning the location of singularities of Taylor-Dirichlet series, on the boundary of convergence.
For a class of sets with multiple terms$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lamb... more For a class of sets with multiple terms$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots,\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1,$$ in weighted \(L^2 (-\infty, \infty)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,...
G. Pólya proved that given a sequence of positive real numbers Λ = { } ∞ =1 of a density and sati... more G. Pólya proved that given a sequence of positive real numbers Λ = { } ∞ =1 of a density and satisfying the gap condition inf ∈N (+1 −) > 0, the Dirichlet series ∑︀ ∞ =1 has at least one singularity in each open interval whose length exceeds 2 and lies on the abscissa of convergence. This raises the question whether the same result holds for a Taylor-Dirichlet series of the form
On a theorem of norman levinson and a variation of the fabry gap theorem
Complex Variables Theory and Application an International Journal, Aug 16, 2006
ABSTRACT In this article we give an extension of a theorem of N. Levinson (see Theorem 2.1). As a... more ABSTRACT In this article we give an extension of a theorem of N. Levinson (see Theorem 2.1). As an application, we get a variation of the Fabry Gap theorem (see Theorem 2.3) concerning the location of singularities of Taylor–Dirichlet series, on the boundary of convergence.
An Addendum to Theorems of A. F. Leont’ev and L. Carleson on an Infinite Order Differential Equation on a Real Interval
Computational Methods and Function Theory, 2012
ABSTRACT In this article we give a necessary and sufficient condition in order for a function to ... more ABSTRACT In this article we give a necessary and sufficient condition in order for a function to be a solution for a special case of an infinite order differential equation that was investigated by L. Carleson and A. F. Leont’ev.
Suppose that {λ n } ∞ n=1 is a sequence of distinct positive real numbers satisfying the conditio... more Suppose that {λ n } ∞ n=1 is a sequence of distinct positive real numbers satisfying the conditions inf{λ n+1 − λ n } > 0, and ∞ n=1 λ −1 n < ∞. We prove that the exponential system {e λnt } ∞ n=1 is hereditarily complete in the closure of the subspace spanned by {e λnt } ∞ n=1 in the space L 2 (a, b). We also give an example of a class of compact non-normal operators defined on this closure which admit spectral synthesis.
We prove that if t u 8 "1 is a complete Riesz-Fischer sequence in a separable Hilbert space , the... more We prove that if t u 8 "1 is a complete Riesz-Fischer sequence in a separable Hilbert space , then :" t P : ÿ |x , y| 2 ă 8u is closed in if and only if t u 8 "1 has a biorthogonal Riesz sequence. If the latter is also complete in , then t u 8 "1 is a Riesz basis for .
Let {e iλnt } n∈Z be an exponential Schauder Basis for L 2 (0, 1), for λ n ∈ R, and let {r n (t)}... more Let {e iλnt } n∈Z be an exponential Schauder Basis for L 2 (0, 1), for λ n ∈ R, and let {r n (t)} n∈Z be its dual Schauder Basis. Let A be a non-empty subset of the integers containing exactly M elements. We prove that for α > 0 the weighted system {t α • r n (t)} n∈Z\A is exact in the space L 2 (0, 1), that is, it is complete and minimal in L 2 (0, 1), if and only if M − 1 2 ≤ α < M + 1 2. We also show that such a system is not a Riesz Basis for L 2 (0, 1). In particular, the weighted trigonometric system {t α • e 2πint } n∈Z\A is exact in L 2 (0, 1), if and only if α ∈ [M − 1 2 , M + 1 2), but it is not a Schauder Basis for L 2 (0, 1).
In this paper we prove that if U = {u n } is a minimal sequence in a separable Hilbert space H , ... more In this paper we prove that if U = {u n } is a minimal sequence in a separable Hilbert space H , then multiplying each vector u n by an appropriate constant c n , yields a family of functions P := {c n • u n } , such that P is a Riesz-Fischer sequence in H. If U is a minimal and complete sequence in H , therefore having a unique biorthogonal sequence V = {v n } in H , then P is a complete Riesz-Fischer sequence in H and its unique biorthogonal family {v n /c n } is a Bessel sequence in H .
Let {λn} ∞ n=1 be a strictly increasing sequence of positive real numbers diverging to infinity a... more Let {λn} ∞ n=1 be a strictly increasing sequence of positive real numbers diverging to infinity and let {μn} ∞ n=1 be a sequence of positive integers. Consider the exponential system EΛ = {t k e λnt : k = 0, 1, 2,. .. , μn − 1} ∞ n=1. Assuming the density condition lim t→∞ λn≤t μn t = d < ∞ and some other restrictions, we prove that every function in the closure of the linear span of EΛ in some weighted Banach spaces on the real line R is extended to an entire function represented by a Taylor-Dirichlet series g(z) = ∞ n=1 μn −1 k=0 c n,k z k e λnz , c n,k ∈ C. We also consider a problem in a weighted L 2 (R) Hilbert space as well as a moment problem on the real line.
The closed span of some Exponential system E Λ in the spaces L p (γ, β), properties of a Biorthog... more The closed span of some Exponential system E Λ in the spaces L p (γ, β), properties of a Biorthogonal family to E Λ in L 2 (γ, β), Moment problems, and a differential equation of Carleson
In this article we give new criteria for two complex sequences to have the same excess in the sen... more In this article we give new criteria for two complex sequences to have the same excess in the sense of Paley and Wiener in L 2 (−a, a). As a result, we prove that given any positive integer q, a real
Perturbation of zeros of entire functions of exponential type
This thesis is divided into two parts. In the first part, we give new criteria for two complex se... more This thesis is divided into two parts. In the first part, we give new criteria for two complex sequences to have the same excess in the sense of Paley and Wiener in L2(−a, a). As a result, we prove that given any positive integer q, a real number α ∈ ( 0, 1 2π ) and complex numbers ν0 = 0, νn = nq + iα log |nq|, |n| ≥ 1, the exponential system {tkeitνn : k = 0, 1, . . . , q − 1}n=−∞ has excess 0 in L2(−π, π). In the second part of the thesis, we give an extension of a theorem of N. Levinson (see Theorems 3.1 and 3.2). As an application, we get a variation of the Fabry Gap Theorem for frequencies with finite upper density (see Theorems 3.3 and 3.4), concerning the location of singularities of Taylor-Dirichlet series, on the boundary of convergence.
For a class of sets with multiple terms$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lamb... more For a class of sets with multiple terms$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots,\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1,$$ in weighted \(L^2 (-\infty, \infty)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,...
G. Pólya proved that given a sequence of positive real numbers Λ = { } ∞ =1 of a density and sati... more G. Pólya proved that given a sequence of positive real numbers Λ = { } ∞ =1 of a density and satisfying the gap condition inf ∈N (+1 −) > 0, the Dirichlet series ∑︀ ∞ =1 has at least one singularity in each open interval whose length exceeds 2 and lies on the abscissa of convergence. This raises the question whether the same result holds for a Taylor-Dirichlet series of the form
On a theorem of norman levinson and a variation of the fabry gap theorem
Complex Variables Theory and Application an International Journal, Aug 16, 2006
ABSTRACT In this article we give an extension of a theorem of N. Levinson (see Theorem 2.1). As a... more ABSTRACT In this article we give an extension of a theorem of N. Levinson (see Theorem 2.1). As an application, we get a variation of the Fabry Gap theorem (see Theorem 2.3) concerning the location of singularities of Taylor–Dirichlet series, on the boundary of convergence.
An Addendum to Theorems of A. F. Leont’ev and L. Carleson on an Infinite Order Differential Equation on a Real Interval
Computational Methods and Function Theory, 2012
ABSTRACT In this article we give a necessary and sufficient condition in order for a function to ... more ABSTRACT In this article we give a necessary and sufficient condition in order for a function to be a solution for a special case of an infinite order differential equation that was investigated by L. Carleson and A. F. Leont’ev.
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Papers by Elias Zikkos