Министерство образования и науки РФ Санкт-Петербургское отделение математического института им. В.А. Стеклова РАН Воронежский государственный университет International Conference WAVELETS AND APPLICATIONS June 14-20. 2009, St.Petersburg, Russia ABSTRACTS Санкт-Петербург 2009 УДК 517.5 Международная конференция "Всплески и приложения, 14-20 июня 2009 г. Санкт-Петербург, Россия". Тезисы докладов ОРГКОМИТЕТ Председатель: И.Я. Новиков (проф. математического факультета Воронежского госуниверситета) Ученый секретарь: П.Г. Северов (преподаватель математи- ческого факультета Воронежского госуниверситета) Члены оргкомитета: Л.В. Новиков (ведущий науч. сотрудник Института аналитического приборостроения РАН) В.В. Капустин (старший науч. сотрудник лаборатории математического анализа Санкт-Петербургского отделения математического института им. В.А.Стеклова) М.А. Скопина (проф. факультета прикладной математики - процессов управления Санкт-Петербургского госуниверситета) С.А. Гарьковская (аспирант математического факультета Воронежского госуниверситета) Издание осуществлено при поддержке Российского фонда фун- даментальных исследований (проект № 09-01-06036-г) Компьютерная верстка и подготовка оригинал-макета: Северов П.Г. WAVELETS AND APPLICATIONS 3 CONTINUOUS WAVELET TRANSFORM ON THE ONE-SHEETED HYPERBOLOID A. Askari Hemmat, Z. Yazdani Fard Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran [email protected] 1;1 2 2 The one-sheeted hyperboloid H with Cartesian equation x0 ¡ x1 ¡ 2 x2 = ¡1 may be parametrized as x = (sinh Â; cosh  cos '; cosh  sin '); where  2 R; 0 · ' < 2¼: The motions on H1;1 are of two types: (i) rotations: x(Â; ') 7! (Â; ' + '0); and (ii) displacements: x(Â; ') 7! (Â+Â0;'): They constitute the group SO0(1; 2). To define 1;1 dilation on H , project it onto the cone C = f» = (»0;»1;»2) 2 3 2 2 2 R : »0 ¡ »1 ¡ »2 = 0g; with dilatin » 7! a»: In polar coordinates, the dilation oprator acts on a point x(Â; ') by Da(Â; ') = (Âa;'), with sinhÂa = asinhÂ: For all test functions f on H1;1 introduce the following pair of transforms ^ § + f§(º; ») = hf(x);"º; »(x)i; where º 2 R ;» varies on the half cone + § C = f» 2 C : »0 > 0g and the kernels "º; »(x) are hyperbolic plane waves. This transformation is called the Fourier-Helgason transform. Let à 2 L2(H1;1) be a symmetric and rotation invariant function, + a 7! ®(a) a positive function on R¤ , we say that à is admissible if there exists constants m and M such that0 < m ·AÃ(º) := Z 1 c 2 1;1 j(Ãa)§(º)j ®(a)da · M < 1: Given an admissible H -wavelet Ã, 0 1;1 2 1;1 the H -continuousZ wavelet transform of f 2 L (H ) is Wf (a; g) := ¡1 hÃa; g; fi = Ãa(g x)f(x)d¹(x); g 2 SO0(1; 2); a > 0 where H1;1 1 Ãa(x) = ¸(a; x) 2 Ã(D 1 x) such that ¸(a; x) is the Radon-Nikodym a derivative. We show that a symmetric and rotation invariant function à 2 L2(H1;1) is admissible if ®(a)da be a homogeneous maesure of ¡¯ the formZ® da with ¯ > 3 and the following zero-mean condition is satisfied Ã(Â; ')d¹(Â; ') = 0: H1;1 References 1. S.T. Ali, J-P. Antoine, and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, Berlin, 4 International Conference Heidelberg, 2000. 2. J. Bros and U. Moschella, Fourier analysis and holomorphic decomposition on the one-sheeted hyperboloid, (2003) [arXiv: math-ph/0311052v1]. DECONVOLUTION BY MATCHING PURSUIT USING SPLINE WAVELET PACKETS DICTIONARIES Amir Z. Averbuch, Valery A. Zheludev School of Computer Science, Tel Aviv University, Israel [email protected] We present an efficient method that restores signals from strongly noised blurred discrete data. The method can be characterized as a regularized matching pursuit (MP), where dictionaries consist of spline wavelet packets. It combines ideas from spline theory, wavelet analysis, theory of ill-posed problems and greedy algorithms. The computational engine, which enables to construct versatile libraries of spline wavelet packet dictionaries and fast implementation of the algorithm, is the Spline Harmonic Analysis (SHA). SHA imposes harmonic analysis methodology onto spline spaces. It is especially applicable to convolution operations. The use of splines enables to map the discrete noised data into spaces of continuous functions, which approximate the sought after solution in the proper smoothed class. The main distinction from the conventional MP is that the different dictionaries are used to test data and to approximate the solution. In addition, the regularized correlation coefficients instead of the conventional ones are used. The regularizing parameters and the stopping rule for the algorithm are determined automatically. Experimental results exhibit a highly efficient algorithm. The coherent structure of the signals, which were subjected to the strong blurring and immersed into deep noise, were extracted. WAVELETS AND APPLICATIONS 5 AN HYBRID ALGORITHM FOR DATA COMPRESSION Amir Z. Averbuch1, Valery A. Zheludev1, Moshe Guttmann1, Dan D. Kosloff2 1School of Computer Science Tel Aviv University, Israel 2Department of Earth and Planetary Sciences Tel Aviv University, Israel [email protected] We present an algorithm that compresses two-dimensional data arrays, which are piece-wise smooth in one direction and have oscillating events in the other direction. Seismic, hyper-spectral and fingerprints data have this mixed structure. The transform part of the compression process is an algorithm that combines wavelet and the local cosine transform (LCT). The quantization and the entropy coding parts of the compression were taken from the SPIHT codec. To efficiently apply the SPIHT codec to a mixed coefficients array, reordering of the LCT coefficients takes place. This algorithm outperforms other algorithms that are based only on the 2D wavelets transforms. Its compression capabilities are also demonstrated on multimedia images that have a fine texture. The wavelet part in the mixed transform of the hybrid algorithm utilizes the library of Butterworth wavelet transforms. TO ONE ADDITIONAL EXTREMAL PROPERTY OF REGULAR SIMPLEX V.F. Babenko1, Yu.V. Babenko2, N.V. Parfinovych1, D.S. Skorokhodov1 1 Dnepropetrovsk National University, UKRAINE, 2 Sam Houston State University, Huntsville, TX, USA [email protected], [email protected], nparfi[email protected], [email protected] For a d-dimensional simplex T , let Lp(T ), 1 · p · 1, be the space of functions f : T! R endowed with the usual norm k ¢ kLp(T ). Denote d g§(x) = maxfg(x); 0g, x 2 R . Let ®; ¯ > 0 be given. For f 2 Lp(T ), the asymmetric Lp-norm is defined as follows kfkLp;®;¯(T ) := k®f+ + ¯f¡kLp(T ): 6 International Conference t d Let S1(T ) := fg(x) = ax + c : a 2 R ; c 2 R; x 2 T g. For f 2 Lp(T ), set E(f; S1(T ))Lp;®;¯(T ) := inffkf ¡ ukLp;®;¯(T ) : u 2 S1(T )g: Note that E(f; S1(T ))Lp;1;1(T ) is the usual best approximation of f by d P 2 linear functions on T in the Lp-metric. Let Q(x) = xj . We solve the j=1 following extremal problem: E(Q; S1(T ))Lp;®;¯(T ) ! inf; jT j = 1; (1) where jT j stands for the volume of simplex T . To the best of our knowledge, problem (1) was solved only in the case of one-sided approximation (interpolation) and in the case ® = ¯ = 1, p = 2 and d = 2. Problem (1) is important for many questions in Conves Geometry, approximation of convex bodies by polytopes and approximation of surfaces by splines. Let T0 be the regular simplex of unit volume in Rd. We obtained the following result. Theorem. Let ®; ¯ > 0, d 2 N and 1 · p · 1. Then E(Q; S1(T0))L (T ) = inf E(Q; S1(T ))L (T ): p;®;¯ 0 T : jT j=1 p;®;¯ EXACT ASYMPTOTICS OF THE OPTIMAL Lp-ERROR OF ASYMMETRIC LINEAR SPLINE APPROXIMATION V.F. Babenko1, Yu.V. Babenko2, N.V. Parfinovych1, D.S. Skorokhodov1 1 Dnepropetrovsk National University, UKRAINE, 2 Sam Houston State University, Huntsville, TX, USA [email protected], [email protected], nparfi[email protected], [email protected] 2 Let D = [0; 1] . By Lp, 1 · p · 1, denote the space of functions f : D ! R endowed with the usual norm k ¢ kp. Let ® and ¯ be positive continuous on D functions, and let f 2 Lp. Then, asymmetric Lp-norm is defined as follows kfkLp;®;¯(D) := k®f+ + ¯f¡kp; WAVELETS AND APPLICATIONS 7 where g§(¢) = maxfg(¢); 0g. N A collection 4N = 4N (D) = fTigi=1 of N triangles, N 2 N, in the plane is called a triangulation of a set D provided that: any pair of triangles from 4N intersects at most at a common vertex or along a SN common edge, and D = Ti. i=1 Given a triangulation 4N , let S(4N ) be the space of continuous on D functions, which are linear on every triangle Ti 2 4N . Now set RN (f; Lp;®;¯) := inf inf kf ¡ skLp;®;¯(D); 4N s2S(4N ) where the first inf is taken over all triangulations with N triangles. 2 Let T0 be the equilateral triangle. For f 2 C (D), set H(f; ¢) := ³ ´2 @2f @2f @2f @x2 (¢) @y2 (¢) ¡ @xy (¢) . Let 2 2 Cp;®;¯(x; y) := inf ku + v ¡ au ¡ bv ¡ ckL (T ): a;b;c2R p;®(x;y);¯(x;y) 0 Theorem. Let f 2 C2(D); H(f; x; y) ¸ C+ > 0 for all (x; y) 2 D. Let positive continuous on D functions ® and ¯ also be given. Then for all 1 · p < 1, µZ ¶ p+1 p p p 2(p+1) p+1 lim N ¢ RN (f; Lp;®;¯) = H (f; x; y)Cp;®;¯(x; y) dxdy : N!1 D ESTIMATES OF MODULUS OF CONTINUITY FOR FUNCTION FROM BESOV-TYPE SPACE Eugeny Berezhnoi Yaroslavl State University, Yaroslavl, Russia [email protected] Let X be a symmetric function space on I = [0; 1]n, Ã(X; t) = kÂ([0; t1=n]n)jXk - is a fundamental function of X and !(f; h; X) = sup kf(: + ±) ¡ f(:)jXk 0·j±j·h 8 International Conference ® is a modulus of continuity of function f.