Complex analysis and convolution operators

Krivosheev, A. S.; Напалков, В. В.

doi:10.1070/rm1992v047n06abeh000954

Cited by 25 publications

(10 citation statements)

References 36 publications

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“…Statements 3 and 4 are proved much as the corresponding statements for H(D); see [10]. We have already established the prerequisites, i.e., the reflexivity of H(D, U ) and its invariance under differentiation (the latter is a consequence of the Banach completeness theorem).…”

Section: The Elementary Solutions Are Complete In the Space Of Solutimentioning

confidence: 99%

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Spectral synthesis in the kernel of a convolution operator on weighted spaces

Yulmukhametov

2010

St. Petersburg Math. J.

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Section: The Elementary Solutions Are Complete In the Space Of Solutimentioning

confidence: 99%

“…In essence, the proof is based on a theorem in [11], Lemma 2.2 in [10], and the Hörmander theorem on L 2 -estimates. We start with formulating these statements in a form suitable for the present setting.…”

Section: Theorem 2 Let V(λ) Be a Psh Function On C P Satisfying (R)mentioning

confidence: 99%

Spectral synthesis in the kernel of a convolution operator on weighted spaces

Yulmukhametov

2010

St. Petersburg Math. J.

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“…In this connection, it should be noted that, in the case of one variable, the spectral synthesis problem was resolved completely in [4] and [5]. In the case of several variables, it is known that the space of solutions of a homogeneous convolution equation in an arbitrary convex domain admits spectral synthesis; see [6,7].…”

Section: ψ(Ty) T Dσ(y)mentioning

confidence: 99%

“…In the case of several variables, the question has been treated in [27]- [31], [7,24,8] and elsewhere. For n > 1, only certain conditions sufficient for continuability have been obtained.…”

Section: ψ(Ty) T Dσ(y)mentioning

confidence: 99%

Criterion of analytic continuability of functions in principal invariant subspaces on convex domains in $\mathbb{C}^{n}$

Krivosheev¹

2011

St. Petersburg Math. J.

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Abstract. Subspaces invariant under differentiation are studied for spaces of functions analytic on domains of a many-dimensional complex space. For a wide class of domains (in particular, for arbitrary bounded convex domains), a criterion of analytic continuability is obtained for functions in arbitrary nontrivial closed principal invariant subspaces admitting spectral synthesis. §1. Notions and notationFor the reader's convenience, here we collect the main notions used in the paper and fix the notation for them.C n is the n-dimensional complex space. S is the sphere of radius 1 and centered at the origin. B(z, δ) is the ball centered at z = (z 1 , . . . , z n ) and of radius δ. {z k } is a sequence of points in C n . The coordinates z = (z 1 , . . . , z n ) of a vector z are denoted similarly, but the specific meaning of the symbol {z k } will be clear from the context each time.For ς ∈ S and a ≥ 0, we denote by Φ a (ς) the set of sequences {z k } ⊂ C n with the following properties:Passing to a subsequence, on each sequence {z k } ∈ Φ a (ς) we can also impose the additional condition saying that {|z k |} is monotone decreasing. Indeed, put k 1 = 1, and for j ≥ 2 choose k j to be the minimal k with |z k

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“…For n = 1, i.e., for one convolution equation, there are numerous publications devoted to this solvability problem; a complete solution was presented in [3,4] for the case of convex domains in the complex space of arbitrary dimension. For smooth domains, full solution of a system of nonhomogeneous convolution equations was proposed in the paper [5], where there is also a list of the most important references on the topic.…”

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confidence: 99%