Yuri Farkov - Academia.edu (original) (raw)
Papers by Yuri Farkov
International Journal of Wavelets, Multiresolution and Information Processing, 2015
ABSTRACT In this paper, we describe an approach to construct of biorthogonal N-periodic discrete ... more ABSTRACT In this paper, we describe an approach to construct of biorthogonal N-periodic discrete MRA wavelets on the basis of generalized Walsh functions. The proposed method is illustrated by a numerical example.
Mathematical Notes, 2014
ABSTRACT Wavelet expansions in L p -spaces on the locally compact Cantor group G are studied. An ... more ABSTRACT Wavelet expansions in L p -spaces on the locally compact Cantor group G are studied. An order-sharp estimate of the wavelet approximation of an arbitrary function f ∈ L p (G) for 1 ≤ p < ∞ in terms of the modulus of continuity of this function is obtained, and a Jackson-Bernstein type theorem on approximation by wavelets of functions from the class Lip(p)(α; G) is proved.
Facta universitatis - series: Electronics and Energetics, 2008
This paper gives a review of multiresolution analysis and compactly supported orthogonal wavelets... more This paper gives a review of multiresolution analysis and compactly supported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of "integer shifts" of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L 2 -spaces on Vilenkin groups. Several examples are provided to illustrate these results.
Russian Mathematics, 2012
Аннотация. С помощью ядер типа Дирихле-Уолша первым автором недавно были определены периодические... more Аннотация. С помощью ядер типа Дирихле-Уолша первым автором недавно были определены периодические диадические всплески на положительной полупрямой, аналогичные тригонометрическим всплескам Чуи-Маскара. В данной статье излагается обобщение этой конструкции и приведены примеры использования диадических периодических всплесков для кодирования фрактальных функций Римана, Вейерштрасса, Шварца, Ван-дер-Вардена, Ганкеля и Такаги. Ключевые слова: периодические диадические всплески, функции Уолша, ядро Дирихле-Уолша, дискретное преобразование Уолша, кодирование сигналов, фрактальные функции. УДК: 519.677 Ю.А. Фарков профессор, заведующий кафедрой высшей математики, Российский государственный геологоразведочный университет, ул. Миклухо-Маклая, д. 23, г.
P-Adic Numbers, Ultrametric Analysis, and Applications, 2011
ABSTRACT Using the Walsh-Dirichlet type kernel, we construct periodic wavelets on the p-adic Vile... more ABSTRACT Using the Walsh-Dirichlet type kernel, we construct periodic wavelets on the p-adic Vilenkin group. These wavelets are similar to the trigonometric wavelets which were introduced by C. K. Chui and H. N. Mhaskar [1]. Results on the corresponding fast algorithms for decomposition and reconstruction are also discussed.
American Journal of Computational Mathematics, 2012
The main aim of this paper is to present a review of periodic wavelets related to the generalized... more The main aim of this paper is to present a review of periodic wavelets related to the generalized Walsh functions on the p-adic Vilenkin group G p . In addition, we consider several examples of wavelets in the spaces of periodic complex sequences. The case p = 2 corresponds to periodic wavelets associated with the classical Walsh functions.
International Journal of Wavelets, Multiresolution and Information Processing, 2015
An explicit description of all Walsh polynomials generating tight wavelet frames is given. An alg... more An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.
In this paper we prove that wavelet expansions on the Cantor dyadic group G converge unconditiana... more In this paper we prove that wavelet expansions on the Cantor dyadic group G converge unconditianally in the dyadic Hardy space H1 (G). We will do it for wavelets satisfying the regularity condition of Hölder-Lipshitz type.
P-Adic Numbers, Ultrametric Analysis, and Applications, 2011
In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported w... more In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.
Siberian Mathematical Journal, 1981
The basic systems introduced by V. D. Erokhin have been used for the computation of the asymptoti... more The basic systems introduced by V. D. Erokhin have been used for the computation of the asymptotic of the e-entropy and for the estimation of the widths of certain classes of analytic functions.
Russian Mathematics, 2011
Аннотация. В настоящей работе для пространств комплексных периодических последовательностей с пом... more Аннотация. В настоящей работе для пространств комплексных периодических последовательностей с помощью дискретного преобразования Уолша построены аналоги ортогональных и биртогональных вейвлетов, изучавшихся ранее для группы Кантора. Проведенные вычислительные эксперименты демонстрируют эффективность методов обработки изображений, основанных на построенных дискретных вейвлетах.
Mathematical Notes, 2011
In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to constr... more In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to construct new orthogonal wavelet bases defined by finite collections of parameters. Earlier similar bases were defined for the Cantor and Vilenkin groups on the basis of generalized Walsh functions. It is noted that similar constructions can be realized for biorthogonal wavelets as well as for the space 2 (Z + ).
Mathematical Notes, 2009
Suppose that ω(ϕ, · ) is the dyadic modulus of continuity of a compactly supported function ϕ in ... more Suppose that ω(ϕ, · ) is the dyadic modulus of continuity of a compactly supported function ϕ in L 2 (R + ) satisfying a scaling equation with 2 n coefficients. Denote by α ϕ the supremum for values of α > 0 such that the inequality ω(ϕ, 2 −j ) ≤ C2 −αj holds for all j ∈ N. For the cases n = 3 and n = 4, we study the scaling functions ϕ generating multiresolution analyses in L 2 (R + ) and the exact values of α ϕ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in L 2 (R + ) corresponding to the scaling function ϕ coincides with α ϕ .
Mathematical Notes, 2007
We describe a method for constructing compactly supported orthogonal wavelets on a locally compac... more We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with p n numerical coefficients generate multiresolution analyses in L 2 (G). It is noted that the coefficients of these scaling equations can be calculated from the given values of p n parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L 2 (G) is stable and has a linearly independent system of "integer" shifts. We present several examples illustrating these results.
International Journal of Wavelets, Multiresolution and Information Processing, 2015
ABSTRACT In this paper, we describe an approach to construct of biorthogonal N-periodic discrete ... more ABSTRACT In this paper, we describe an approach to construct of biorthogonal N-periodic discrete MRA wavelets on the basis of generalized Walsh functions. The proposed method is illustrated by a numerical example.
Mathematical Notes, 2014
ABSTRACT Wavelet expansions in L p -spaces on the locally compact Cantor group G are studied. An ... more ABSTRACT Wavelet expansions in L p -spaces on the locally compact Cantor group G are studied. An order-sharp estimate of the wavelet approximation of an arbitrary function f ∈ L p (G) for 1 ≤ p < ∞ in terms of the modulus of continuity of this function is obtained, and a Jackson-Bernstein type theorem on approximation by wavelets of functions from the class Lip(p)(α; G) is proved.
Facta universitatis - series: Electronics and Energetics, 2008
This paper gives a review of multiresolution analysis and compactly supported orthogonal wavelets... more This paper gives a review of multiresolution analysis and compactly supported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of "integer shifts" of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L 2 -spaces on Vilenkin groups. Several examples are provided to illustrate these results.
Russian Mathematics, 2012
Аннотация. С помощью ядер типа Дирихле-Уолша первым автором недавно были определены периодические... more Аннотация. С помощью ядер типа Дирихле-Уолша первым автором недавно были определены периодические диадические всплески на положительной полупрямой, аналогичные тригонометрическим всплескам Чуи-Маскара. В данной статье излагается обобщение этой конструкции и приведены примеры использования диадических периодических всплесков для кодирования фрактальных функций Римана, Вейерштрасса, Шварца, Ван-дер-Вардена, Ганкеля и Такаги. Ключевые слова: периодические диадические всплески, функции Уолша, ядро Дирихле-Уолша, дискретное преобразование Уолша, кодирование сигналов, фрактальные функции. УДК: 519.677 Ю.А. Фарков профессор, заведующий кафедрой высшей математики, Российский государственный геологоразведочный университет, ул. Миклухо-Маклая, д. 23, г.
P-Adic Numbers, Ultrametric Analysis, and Applications, 2011
ABSTRACT Using the Walsh-Dirichlet type kernel, we construct periodic wavelets on the p-adic Vile... more ABSTRACT Using the Walsh-Dirichlet type kernel, we construct periodic wavelets on the p-adic Vilenkin group. These wavelets are similar to the trigonometric wavelets which were introduced by C. K. Chui and H. N. Mhaskar [1]. Results on the corresponding fast algorithms for decomposition and reconstruction are also discussed.
American Journal of Computational Mathematics, 2012
The main aim of this paper is to present a review of periodic wavelets related to the generalized... more The main aim of this paper is to present a review of periodic wavelets related to the generalized Walsh functions on the p-adic Vilenkin group G p . In addition, we consider several examples of wavelets in the spaces of periodic complex sequences. The case p = 2 corresponds to periodic wavelets associated with the classical Walsh functions.
International Journal of Wavelets, Multiresolution and Information Processing, 2015
An explicit description of all Walsh polynomials generating tight wavelet frames is given. An alg... more An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.
In this paper we prove that wavelet expansions on the Cantor dyadic group G converge unconditiana... more In this paper we prove that wavelet expansions on the Cantor dyadic group G converge unconditianally in the dyadic Hardy space H1 (G). We will do it for wavelets satisfying the regularity condition of Hölder-Lipshitz type.
P-Adic Numbers, Ultrametric Analysis, and Applications, 2011
In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported w... more In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.
Siberian Mathematical Journal, 1981
The basic systems introduced by V. D. Erokhin have been used for the computation of the asymptoti... more The basic systems introduced by V. D. Erokhin have been used for the computation of the asymptotic of the e-entropy and for the estimation of the widths of certain classes of analytic functions.
Russian Mathematics, 2011
Аннотация. В настоящей работе для пространств комплексных периодических последовательностей с пом... more Аннотация. В настоящей работе для пространств комплексных периодических последовательностей с помощью дискретного преобразования Уолша построены аналоги ортогональных и биртогональных вейвлетов, изучавшихся ранее для группы Кантора. Проведенные вычислительные эксперименты демонстрируют эффективность методов обработки изображений, основанных на построенных дискретных вейвлетах.
Mathematical Notes, 2011
In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to constr... more In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to construct new orthogonal wavelet bases defined by finite collections of parameters. Earlier similar bases were defined for the Cantor and Vilenkin groups on the basis of generalized Walsh functions. It is noted that similar constructions can be realized for biorthogonal wavelets as well as for the space 2 (Z + ).
Mathematical Notes, 2009
Suppose that ω(ϕ, · ) is the dyadic modulus of continuity of a compactly supported function ϕ in ... more Suppose that ω(ϕ, · ) is the dyadic modulus of continuity of a compactly supported function ϕ in L 2 (R + ) satisfying a scaling equation with 2 n coefficients. Denote by α ϕ the supremum for values of α > 0 such that the inequality ω(ϕ, 2 −j ) ≤ C2 −αj holds for all j ∈ N. For the cases n = 3 and n = 4, we study the scaling functions ϕ generating multiresolution analyses in L 2 (R + ) and the exact values of α ϕ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in L 2 (R + ) corresponding to the scaling function ϕ coincides with α ϕ .
Mathematical Notes, 2007
We describe a method for constructing compactly supported orthogonal wavelets on a locally compac... more We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with p n numerical coefficients generate multiresolution analyses in L 2 (G). It is noted that the coefficients of these scaling equations can be calculated from the given values of p n parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L 2 (G) is stable and has a linearly independent system of "integer" shifts. We present several examples illustrating these results.
Пленарный доклад на 18-й Саратовской зимней школе "Современные проблемы теории функций и их прило... more Пленарный доклад на 18-й Саратовской зимней школе
"Современные проблемы теории функций и их приложения"
(27 января - 3 февраля 2016 года)