Yuri Farkov - Academia.edu (original) (raw)

Papers by Yuri Farkov

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.

We describe two type of wavelet tight frames associated with the generalized Walsh functions: (1)... more We describe two type of wavelet tight frames associated with the generalized Walsh functions: (1) Parseval frames for L-spaces on Vilenkin groups, (2) finite tight frames for the space `(ZN ). In particular cases these tight frames coincide with orthogonal wavelet bases associated with the classical Walsh functions.

Известия Российской академии наук. Серия математическая, 2005

Matematicheskie Zametki, 2007

Излагается метод построения ортогональных вейвлетов с компактными носителями на локально компактн... more Излагается метод построения ортогональных вейвлетов с компактными носителями на локально компактной абелевой группе G, являющейся слабым прямым произведением счетного множества циклических групп p-го порядка. Для любых целых p, n 2 установлены необходимые и достаточные условия для того, чтобы решения соответствующих масштабирующих уравнений с p n числовыми коэффициентами генерировали кратномасштабные анализы в L 2 (G). Отмечается, что коэффициенты этих масштабирующих уравнений вычисляются по заданным значениям p n параметров с помощью дискретного преобразования Виленкина-Крестенсона. Кроме того, найдены условия, при которых финитное решение масштабирующего уравнения в L 2 (G) стабильно и имеет линейно независимую систему "целочисленных" сдвигов. Приведено несколько примеров, иллюстрирующих эти результаты. Библиография: 16 названий.

2017 International Conference on Sampling Theory and Applications (SampTA)

It is well known that generalized Walsh functions can be considered as characters of Vilenkin gro... more It is well known that generalized Walsh functions can be considered as characters of Vilenkin groups (see, e.g., [6], [11]). For wavelets on Vilenkin groups most of the results relate to the locally compact group G<inf>p</inf>, which is defined by a fixed integer p ≥ 2 (the case p = 2 corresponds to the Cantor group). In this paper, we are interested in an nonstationary MRA and related wavelets for the space L²(G<inf>p</inf>); for the stationary case see [2], [4], [5], and references therein. Furthermore, we give an algorithm for construction of nonstationary wavelets on the Vilenkin group associated with a sequence {p<inf>j</inf>}<inf>j=1</inf>^∞, p<inf>j</inf> ≥ 2, of natural numbers.

Communications in Mathematics and Applications

Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of ... more Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of frames and periodic wavelets on the positive halfline. The corresponding algorithms for decomposition and reconstruction are also discussed. It is noted that similar results can be obtained for wavelets and frames on the Cantor and Vilenkin groups.

Communications in Mathematics and Applications, 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 Gp. 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.

European Journal of Mathematics, 2018

We describe three types of compactly supported wavelet frames associated with Walsh functions: (1... more We describe three types of compactly supported wavelet frames associated with Walsh functions: (1) MRA-based tight frames, (2) frames obtained from the Daubechies-type "admissible condition", and (3) frames based on the Walsh-Parseval type kernels. Parametric wavelet sets for Vilenkin groups and some related results are also discussed.

Computer Research and Modeling, 2017

The article proposes a method of joint analysis of multidimensional financial time series based o... more The article proposes a method of joint analysis of multidimensional financial time series based on the evaluation of the set of properties of stock quotes in a sliding time window and the subsequent averaging of property values for all analyzed companies. The main purpose of the analysis is to construct measures of joint behavior of time series reacting to the occurrence of a synchronous or coherent component. The coherence of the behavior of the characteristics of a complex system is an important feature that makes it possible to evaluate the approach of the system to sharp changes in its state. The basis for the search for precursors of sharp changes is the general idea of increasing the correlation of random fluctuations of the system parameters as it approaches the critical state. The increments in time series of stock values have a pronounced chaotic character and have a large amplitude of individual noises, against which a weak common signal can be detected only on the basis of its correlation in different scalar components of a multidimensional time series. It is known that classical methods of analysis based on the use of correlations between neighboring samples are ineffective in the processing of financial time series, since from the point of view of the correlation theory of random processes, increments in the value of shares formally have all the attributes of white noise (in particular, the "flat spectrum" and "delta-shaped" autocorrelation function). In connection with this, it is proposed to go from analyzing the initial signals to examining the sequences of their nonlinear properties calculated in time fragments of small length. As such properties, the entropy of the wavelet coefficients is used in the decomposition into the Daubechies basis, the multifractal parameters and the autoregressive measure of signal nonstationarity. Measures of synchronous behavior of time series properties in a sliding time window are constructed using the principal component method, moduli values of all pairwise correlation coefficients, and a multiple spectral coherence measure that is a generalization of the quadratic coherence spectrum between two signals. The shares of 16 large Russian companies from the beginning of 2010 to the end of 2016 were studied. Using the proposed method, two synchronization time intervals of the Russian stock market were identified: from mid-December 2013 to mid

Journal of Mathematical Sciences, 2016

Let Ω be an open subset of the complex plane C and let E be a compact subset of Ω. The present su... more Let Ω be an open subset of the complex plane C and let E be a compact subset of Ω. The present survey is concerned with linear n-widths for the class H ∞ (Ω) in the space C(E) and some problems on the best linear approximation of classes of Hardy-Sobolev-type in L p-spaces. It is known that the partial sums of the Faber series give the classical method for approximation of functions f ∈ H ∞ (Ω) in the metric of C(E) when E is a bounded continuum with simply connected complement and Ω is a canonical neighborhood of E. Generalizations of the Faber series are defined for the case where Ω is a multiply connected domain or a disjoint union of several such domains, while E can be split into a finite number of continua. The exact values of n-widths and asymptotic formulas for the ε-entropy of classes of holomorphic functions with bounded fractional derivatives in domains of tube type are presented. Also, some results about Faber's approximations in connection with their applications in numerical analysis are mentioned.

International Journal of Wavelets, Multiresolution and Information Processing, 2015

In this paper, we describe an approach to construct of biorthogonal N-periodic discrete MRA wavel... more 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 &lt; ∞ 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 sup- ported orthogonal wavele... more This paper gives a review of multiresolution analysis and compactly sup- ported 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 L2-spaces on Vilenkin groups. Several examples are provided to illustrate these results. .

Russian Mathematics, 2012

Recently, using the Walsh-Dirichlet type kernels, the first author has defined periodic dyadic wa... more Recently, using the Walsh-Dirichlet type kernels, the first author has defined periodic dyadic wavelets on the positive semiaxis which are similar to the Chui-Mhaskar trigonometric wavelets. In this paper we generalize this construction and give examples of applications of periodic dyadic wavelets for coding the Riemann, Weierstrass, Schwarz, van der Waerden, Hankel, and Takagi fractal functions.

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

Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dy... more Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

Communications in Mathematics and Applications

Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of ... more Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of frames and periodic wavelets on the positive halfline. The corresponding algorithms for decomposition and reconstruction are also discussed. It is noted that similar results can be obtained for wavelets and frames on the Cantor and Vilenkin groups.

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.

We describe two type of wavelet tight frames associated with the generalized Walsh functions: (1)... more We describe two type of wavelet tight frames associated with the generalized Walsh functions: (1) Parseval frames for L-spaces on Vilenkin groups, (2) finite tight frames for the space `(ZN ). In particular cases these tight frames coincide with orthogonal wavelet bases associated with the classical Walsh functions.

Известия Российской академии наук. Серия математическая, 2005

Matematicheskie Zametki, 2007

Излагается метод построения ортогональных вейвлетов с компактными носителями на локально компактн... more Излагается метод построения ортогональных вейвлетов с компактными носителями на локально компактной абелевой группе G, являющейся слабым прямым произведением счетного множества циклических групп p-го порядка. Для любых целых p, n 2 установлены необходимые и достаточные условия для того, чтобы решения соответствующих масштабирующих уравнений с p n числовыми коэффициентами генерировали кратномасштабные анализы в L 2 (G). Отмечается, что коэффициенты этих масштабирующих уравнений вычисляются по заданным значениям p n параметров с помощью дискретного преобразования Виленкина-Крестенсона. Кроме того, найдены условия, при которых финитное решение масштабирующего уравнения в L 2 (G) стабильно и имеет линейно независимую систему "целочисленных" сдвигов. Приведено несколько примеров, иллюстрирующих эти результаты. Библиография: 16 названий.

2017 International Conference on Sampling Theory and Applications (SampTA)

It is well known that generalized Walsh functions can be considered as characters of Vilenkin gro... more It is well known that generalized Walsh functions can be considered as characters of Vilenkin groups (see, e.g., [6], [11]). For wavelets on Vilenkin groups most of the results relate to the locally compact group G<inf>p</inf>, which is defined by a fixed integer p ≥ 2 (the case p = 2 corresponds to the Cantor group). In this paper, we are interested in an nonstationary MRA and related wavelets for the space L²(G<inf>p</inf>); for the stationary case see [2], [4], [5], and references therein. Furthermore, we give an algorithm for construction of nonstationary wavelets on the Vilenkin group associated with a sequence {p<inf>j</inf>}<inf>j=1</inf>^∞, p<inf>j</inf> ≥ 2, of natural numbers.

Communications in Mathematics and Applications

Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of ... more Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of frames and periodic wavelets on the positive halfline. The corresponding algorithms for decomposition and reconstruction are also discussed. It is noted that similar results can be obtained for wavelets and frames on the Cantor and Vilenkin groups.

Communications in Mathematics and Applications, 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 Gp. 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.

European Journal of Mathematics, 2018

We describe three types of compactly supported wavelet frames associated with Walsh functions: (1... more We describe three types of compactly supported wavelet frames associated with Walsh functions: (1) MRA-based tight frames, (2) frames obtained from the Daubechies-type "admissible condition", and (3) frames based on the Walsh-Parseval type kernels. Parametric wavelet sets for Vilenkin groups and some related results are also discussed.

Computer Research and Modeling, 2017

The article proposes a method of joint analysis of multidimensional financial time series based o... more The article proposes a method of joint analysis of multidimensional financial time series based on the evaluation of the set of properties of stock quotes in a sliding time window and the subsequent averaging of property values for all analyzed companies. The main purpose of the analysis is to construct measures of joint behavior of time series reacting to the occurrence of a synchronous or coherent component. The coherence of the behavior of the characteristics of a complex system is an important feature that makes it possible to evaluate the approach of the system to sharp changes in its state. The basis for the search for precursors of sharp changes is the general idea of increasing the correlation of random fluctuations of the system parameters as it approaches the critical state. The increments in time series of stock values have a pronounced chaotic character and have a large amplitude of individual noises, against which a weak common signal can be detected only on the basis of its correlation in different scalar components of a multidimensional time series. It is known that classical methods of analysis based on the use of correlations between neighboring samples are ineffective in the processing of financial time series, since from the point of view of the correlation theory of random processes, increments in the value of shares formally have all the attributes of white noise (in particular, the "flat spectrum" and "delta-shaped" autocorrelation function). In connection with this, it is proposed to go from analyzing the initial signals to examining the sequences of their nonlinear properties calculated in time fragments of small length. As such properties, the entropy of the wavelet coefficients is used in the decomposition into the Daubechies basis, the multifractal parameters and the autoregressive measure of signal nonstationarity. Measures of synchronous behavior of time series properties in a sliding time window are constructed using the principal component method, moduli values of all pairwise correlation coefficients, and a multiple spectral coherence measure that is a generalization of the quadratic coherence spectrum between two signals. The shares of 16 large Russian companies from the beginning of 2010 to the end of 2016 were studied. Using the proposed method, two synchronization time intervals of the Russian stock market were identified: from mid-December 2013 to mid

Journal of Mathematical Sciences, 2016

Let Ω be an open subset of the complex plane C and let E be a compact subset of Ω. The present su... more Let Ω be an open subset of the complex plane C and let E be a compact subset of Ω. The present survey is concerned with linear n-widths for the class H ∞ (Ω) in the space C(E) and some problems on the best linear approximation of classes of Hardy-Sobolev-type in L p-spaces. It is known that the partial sums of the Faber series give the classical method for approximation of functions f ∈ H ∞ (Ω) in the metric of C(E) when E is a bounded continuum with simply connected complement and Ω is a canonical neighborhood of E. Generalizations of the Faber series are defined for the case where Ω is a multiply connected domain or a disjoint union of several such domains, while E can be split into a finite number of continua. The exact values of n-widths and asymptotic formulas for the ε-entropy of classes of holomorphic functions with bounded fractional derivatives in domains of tube type are presented. Also, some results about Faber's approximations in connection with their applications in numerical analysis are mentioned.

International Journal of Wavelets, Multiresolution and Information Processing, 2015

In this paper, we describe an approach to construct of biorthogonal N-periodic discrete MRA wavel... more 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 &lt; ∞ 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 sup- ported orthogonal wavele... more This paper gives a review of multiresolution analysis and compactly sup- ported 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 L2-spaces on Vilenkin groups. Several examples are provided to illustrate these results. .

Russian Mathematics, 2012

Recently, using the Walsh-Dirichlet type kernels, the first author has defined periodic dyadic wa... more Recently, using the Walsh-Dirichlet type kernels, the first author has defined periodic dyadic wavelets on the positive semiaxis which are similar to the Chui-Mhaskar trigonometric wavelets. In this paper we generalize this construction and give examples of applications of periodic dyadic wavelets for coding the Riemann, Weierstrass, Schwarz, van der Waerden, Hankel, and Takagi fractal functions.

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

Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dy... more Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

Communications in Mathematics and Applications

Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of ... more Using the Walsh-Dirichlet kernel and some of its modifications, we construct several examples of frames and periodic wavelets on the positive halfline. The corresponding algorithms for decomposition and reconstruction are also discussed. It is noted that similar results can be obtained for wavelets and frames on the Cantor and Vilenkin groups.