The fundamental principle for invariant subspaces of analytic functions. I

Krasichkov-Ternovskii, Igor Fedorovich

doi:10.1070/sm1997v188n02abeh000200

Cited by 2 publications

(5 citation statements)

References 28 publications

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“…The function 1 ( , ) satisfies estimate (10) and relation (11), and this is why by Lemma 17 and Lemma 5.1 in book [11] we obtain the needed statement. Assume now that the vector function…”

Section: Summation Of Series Of Elementary Solutionsmentioning

confidence: 76%

“…Let us compare Theorem 1 with Theorem 4.4 in [10]), which is the main result of the series of papers [8]- [10] for a square non-degenerate system of convolution equations. and by Theorem 1, for each vector function ∈ ( ) × ( ) solving the considered homogeneous system, the sequence ( , , , ) for some sequence of circumferences converges to this vector function in the topology of the space ( ) × ( ).…”

Section: Examplementioning

confidence: 99%

“…is a pair of convex domains in the complex plane and ′ ⊂ C is a compact set [10]. This means that inequalities hold:…”

Section: Examplementioning

confidence: 99%

“…We consider the system of convolution equations In works by I.F. Krasichkov-Ternovskii [1]- [7], the approximation of an arbitrary solution to this system by linear combinations of elementary solutions was studied, while works [8]- [10] were devoted to the summability of a series of elementary solutions.…”

Section: Introductionmentioning

confidence: 99%

“…In representation(8), the vector function is independent of the parameter , is a linear continuous functional on the space∏︁ =1 ( )w.r.t. the variable , while w.r.t.…”

mentioning

confidence: 99%

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Systems of convolution equations in complex domains

Merzlyakov¹

2018

Ufimskii Matematicheskii Zhurnal

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In this paper we study the systems of convolution equations in spaces of vectorvalued functions of one variable. We define an analogue of the Leontiev interpolating function for such systems, and we provide a series of the properties of this function. In order to study these systems, we introduce a geometric difference of sets and provide its properties. We prove a theorem on the representation of arbitrary vector-valued functions as a series over elementary solutions to the homogeneous system of convolution equations. These results generalize some well-known results by A.F. Leontiev on methods of summing a series of elementary solutions to an arbitrary solution and strengthen the results by I.F. Krasichkov-Ternovskii on summability of a square system of convolution equations. We describe explicitly domains in which a series of elementary solutions converges for arbitrary vector-valued functions. These domains depend on the domains of the vectorvalued functions, on the growth of the Laplace transform of the elements in this system, and on the lower bound of its determinant. We adduce examples showing the sharpness of this result. Similar results are obtained for solutions to a homogeneous system of convolution equations, and examples are given in which the series converges in the entire domain of a vector-valued function.

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“…The function 1 ( , ) satisfies estimate (10) and relation (11), and this is why by Lemma 17 and Lemma 5.1 in book [11] we obtain the needed statement. Assume now that the vector function…”

Section: Summation Of Series Of Elementary Solutionsmentioning

confidence: 76%

Section: Examplementioning

confidence: 99%

“…is a pair of convex domains in the complex plane and ′ ⊂ C is a compact set [10]. This means that inequalities hold:…”

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In representation(8), the vector function is independent of the parameter , is a linear continuous functional on the space∏︁ =1 ( )w.r.t. the variable , while w.r.t.…”

mentioning

confidence: 99%

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Systems of convolution equations in complex domains

Merzlyakov¹

2018

Ufimskii Matematicheskii Zhurnal

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Spectral synthesis for systems of differential operators with constant coefficients

Shishkin¹

2003

Sb. Math.

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Bit data was recorded on a LiTaO 3 single-crystal medium using a data storage system which was based on scanning nonlinear dielectric microscopy, and bit error rate was evaluated. The recording medium with a highly homogeneous thickness of 119 nm was prepared employing both polarization controlled wet etching and dry etching techniques. A 256 Â 256 data bit array was recorded at an areal density of 258 Gbit/inch 2 . The bit error rate was determined to be 1:2 Â 10 À3 by visual inspection. An automated analysis method was subsequently discussed in detail.

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The fundamental principle for invariant subspaces of analytic functions. I

Cited by 2 publications

References 28 publications

Systems of convolution equations in complex domains

Systems of convolution equations in complex domains

Spectral synthesis for systems of differential operators with constant coefficients

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