{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:40:46Z","timestamp":1760240446298,"version":"build-2065373602"},"reference-count":98,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,21]],"date-time":"2019-06-21T00:00:00Z","timestamp":1561075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional H\u00f6lder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional H\u00f6lder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets.<\/jats:p>","DOI":"10.3390\/sym11060825","type":"journal-article","created":{"date-parts":[[2019,6,21]],"date-time":"2019-06-21T11:54:31Z","timestamp":1561118071000},"page":"825","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Directional Thermodynamic Formalism"],"prefix":"10.3390","volume":"11","author":[{"given":"Mourad","family":"Ben Slimane","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Moez","family":"Ben Abid","sequence":"additional","affiliation":[{"name":"Ecole Sup\u00e9rieure des Sciences et de la Technologie de Hammam Sousse, Universit\u00e9 de Sousse, Sousse 4011, Tunisia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ines","family":"Ben Omrane","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Borhen","family":"Halouani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3521","DOI":"10.1088\/0305-4470\/17\/18\/021","article-title":"On the multifractal nature of turbulence and chotic systems","volume":"17","author":"Benzi","year":"1984","journal-title":"J. Phys. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1017\/S0022112074000711","article-title":"Intermittent turbulence in selfsimilar cascades: Divergence of high moments and dimension of the carrier","volume":"62","author":"Mandelbrot","year":"1974","journal-title":"J. Fluid Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1103\/RevModPhys.57.617","article-title":"Ergodic theory of chaos and strange attractors","volume":"57","author":"Eckmann","year":"1985","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1103\/PhysRevA.33.1141","article-title":"Fractal measures and their singularities: The charaterization of strange sets","volume":"33","author":"Halsey","year":"1986","journal-title":"Phys. Rev. A"},{"key":"ref_5","first-page":"16","article-title":"Energy dissipation in locally isotropic turbulence","volume":"32","author":"Kolmogorov","year":"1941","journal-title":"Dokl. Akad. Nauk."},{"key":"ref_6","first-page":"8285","article-title":"A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number","volume":"12","author":"Kolmogorov","year":"1962","journal-title":"J. Fluid Mech."},{"key":"ref_7","first-page":"7781","article-title":"Some specific features of atmospheric tubulence","volume":"12","author":"Oboukhov","year":"1962","journal-title":"J. Fluid Mech."},{"key":"ref_8","unstructured":"Mandelbrot, B. (1975). Les Objets Fractals: Forme, Hasard et Dimension, Flammarian."},{"key":"ref_9","unstructured":"Mandelbrot, B. (1982). The Fractal Geometry of Nature, W. H. Freeman."},{"key":"ref_10","unstructured":"Fermi, E. Fully developped turbulence and intermittency. Proceedings of the International Summer School in Physics."},{"key":"ref_11","first-page":"807","article-title":"Multifractal analysis of measures","volume":"319","year":"1994","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"ref_12","first-page":"264","article-title":"A necessary condition and sufficientn condition for a valid multifractal formalism","volume":"165","author":"Bhouri","year":"2002","journal-title":"Math. Adv. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1007\/BF01055700","article-title":"On the multifractal analysis of measures","volume":"66","author":"Brown","year":"1992","journal-title":"J. Statist. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1007\/BF01206149","article-title":"The dimension specrum of some dynamical systems","volume":"47","author":"Collet","year":"1987","journal-title":"J. Statist. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Falconer, K.-J. (1990). Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons.","DOI":"10.2307\/2532125"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1512\/iumj.1981.30.30055","article-title":"Fractals and self-similarity","volume":"30","author":"Hutchinson","year":"1981","journal-title":"Indiana Univ. Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/aima.1995.1061","article-title":"The singularity spectrum for general Sierpinski carpets","volume":"116","author":"King","year":"1995","journal-title":"Adv. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1006\/aima.1995.1066","article-title":"A multifractal formalism","volume":"116","author":"Olsen","year":"1995","journal-title":"Adv. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"143","DOI":"10.2140\/pjm.1998.183.143","article-title":"Self-affine multifractal Sierpinski sponges in Rd","volume":"183","author":"Olsen","year":"1998","journal-title":"Pac. J. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1017\/S0143385700005162","article-title":"The singularity spectrum f(\u03b1) for cookie-cutters","volume":"9","author":"Rand","year":"1989","journal-title":"Ergodic Theory Dyn. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1142\/S0129055X94000353","article-title":"On the thermodynamic formalism for functions","volume":"6","author":"Daubechies","year":"1994","journal-title":"Rev. Math. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/BF01053588","article-title":"Singularity spectrum of fractal signals from wavelet analysis: Exact results","volume":"70","author":"Arneodo","year":"1993","journal-title":"J. Statist. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/0378-4371(94)00163-N","article-title":"The thermodynamics of fractals revisited with wavelets","volume":"213","author":"Arneodo","year":"1995","journal-title":"Physica A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1142\/S0218127494000204","article-title":"A multifractal formalism revisited with wavelets","volume":"4","author":"Muzy","year":"1994","journal-title":"Internat. J. Bifur. Chaos Appl. Sci. Engrg."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"944","DOI":"10.1137\/S0036141095282991","article-title":"Multifractal formalism for functions. Part 1: Results valid for all functions and Part 2: Selfsimilar functions","volume":"28","author":"Jaffard","year":"1997","journal-title":"SIAM J. Math. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s00365-009-9042-6","article-title":"H\u00f6lder regularity of \u03bc-similar functions","volume":"31","author":"Seuret","year":"2010","journal-title":"Const. Approx."},{"key":"ref_27","unstructured":"Ben Slimane, M. (1996). Etude du Formalisme Multifractal pour les Fonctions. [Ph.D. Thesis, Ecole Nationale des Ponts et Chauss\u00e9es]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1016\/S0764-4442(97)87872-X","article-title":"Formalisme Multifractal pour quelques g\u00e9n\u00e9ralisations des fonctions autosimilaires","volume":"324","year":"1997","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1017\/S0305004198002710","article-title":"Multifractal formalism and anisotropic selfsimilar functions","volume":"124","year":"1998","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_30","first-page":"209","article-title":"Multifractal formalism for selfsimilar functions under the action of nonlinear dynamical systems","volume":"15","year":"1994","journal-title":"Constr. Approx."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1006\/acha.2001.0364","article-title":"Multifractal formalism for selfsimilar functions expanded in singular basis","volume":"11","year":"2001","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s004400050224","article-title":"The multifractal nature of the L\u00e9vy processes","volume":"114","author":"Jaffard","year":"1999","journal-title":"Probab. Theory Related Fields"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1080\/1065246031000074335","article-title":"Some functional equations revisited: The multifractal properties","volume":"14","year":"2003","journal-title":"Integral Transf. Spec. Funct."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"441","DOI":"10.4171\/rmi\/203","article-title":"The spectrum of singularities of Riemann\u2019s function","volume":"12","author":"Jaffard","year":"1996","journal-title":"Rev. Math. Iberoam."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1016\/S0021-7824(00)00161-6","article-title":"On the Frisch-Parisi conjecture","volume":"79","author":"Jaffard","year":"2000","journal-title":"J. Math. Pures Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1137\/060669760","article-title":"Generic validity of the multifractal formalism","volume":"39","author":"Fraysse","year":"2007","journal-title":"SIAM J. Math. Anal. Soc. Ind. Appl. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"663","DOI":"10.4171\/rmi\/469","article-title":"How smooth is almost every function in Sobolev space?","volume":"22","author":"Fraysse","year":"2006","journal-title":"Rev. Math. Iberoam."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"044501","DOI":"10.1103\/PhysRevLett.93.044501","article-title":"Generalizing the wavelet-based multifractal formalism to vector-valued random fields: Application to turbulent velocity and vorticity 3D numerical data","volume":"93","author":"Kestener","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"313","DOI":"10.4171\/rmi\/836","article-title":"Hyperbolic wavelet transform: An efficient tool for multifractal analysis of anisotropic textures","volume":"31","author":"Abry","year":"2015","journal-title":"Rev. Math. Iberoam."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/MSP.2015.2402056","article-title":"Multiscale Anisotropic Texture Analysis and Classification of Photographic Prints: Art scholarship meets image processing algorithms","volume":"32","author":"Abry","year":"2015","journal-title":"IEEE Signal Proc. Mag."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Bunde, A., Kropp, J., and Schellnhuber, H.J. (2002). Wavelet-based multifractal formalism: Applications to DNA sequences, satellite images of the cloud structure and stock market data. The Science of Disasters, Springer.","DOI":"10.1007\/978-3-642-56257-0"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/j.jfa.2012.11.012","article-title":"Local behavior of traces of Besov functions: Prevalent results","volume":"264","author":"Aubry","year":"2013","journal-title":"J. Funct. Anal."},{"key":"ref_43","first-page":"385","article-title":"Directional regularity criteria","volume":"349","year":"2011","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1007\/s00041-012-9226-5","article-title":"Directional and anisotropic regularity and irregularity criteria in Triebel wavelet bases","volume":"18","year":"2012","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1142\/S0218348X12500223","article-title":"Wavelet characterizations of multi-directional regularity","volume":"20","year":"2012","journal-title":"Fractals"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1142\/S0218348X11005208","article-title":"Explicit constructions of operator scaling Gaussian fields","volume":"19","author":"Clausel","year":"2011","journal-title":"Fractals"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1111\/1467-9868.00160","article-title":"Fractal analysis of surface roughness by using spatial data","volume":"61","author":"Davies","year":"1999","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.acha.2010.02.002","article-title":"Pointwise and directional regularity of nonharmonic Fourier series","volume":"28","author":"Jaffard","year":"2010","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10704-005-3059-z","article-title":"Anisotropic self-affine properties of experimental fracture surfaces","volume":"140","author":"Ponson","year":"2006","journal-title":"Int. J. Fracture"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"4353","DOI":"10.1109\/TIP.2013.2272515","article-title":"Self-Similar Anisotropic Texture Analysis: The Hyperbolic Wavelet Transform Contribution","volume":"22","author":"Roux","year":"2013","journal-title":"IEEE Trans. Image Proc."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/s00041-008-9054-9","article-title":"Estimations of H\u00f6lder Regularities and Direction of Singularity by Hart Smith and Curvelet Transforms","volume":"15","author":"Sampo","year":"2009","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_52","unstructured":"Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators, North-Holland."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s00365-012-9166-y","article-title":"Parabolic Besov regularity for the heat equation","volume":"36","author":"Aimar","year":"2012","journal-title":"Constr. Approx."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.spa.2006.07.004","article-title":"Operator scaling stable random fields","volume":"117","author":"Meerschaert","year":"2007","journal-title":"Stoch. Proc. Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s00041-003-0012-2","article-title":"Anisotropic analysis of some Gaussian models","volume":"9","author":"Bonami","year":"2003","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1086\/505144","article-title":"Morphological Analysis of H I Features. II. Wavelet-based multifractal formalism","volume":"165","author":"Khalil","year":"2006","journal-title":"Astrophys. J. Suppl. Ser."},{"key":"ref_57","first-page":"1","article-title":"An adapted group dilation anisotropic multifractal formalism for functions","volume":"15","year":"2008","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1016\/j.jmaa.2017.11.061","article-title":"Criteria of pointwise and uniform directional Lipschitz regularities on tensor products of Schauder functions","volume":"460","author":"Halouani","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s13163-015-0185-7","article-title":"Baire generic results for the anisotropic multifractal formalism","volume":"29","year":"2016","journal-title":"Rev. Mater. Complut."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/s00209-005-0765-1","article-title":"Atomic and molecular decomposition of anisotropic Besov spaces","volume":"250","author":"Bownik","year":"2005","journal-title":"Math. Z."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1469","DOI":"10.1090\/S0002-9947-05-03660-3","article-title":"Atomic and molecular decomposition of anisotropic Triebel- Lizorkin spaces","volume":"385","author":"Bownik","year":"2005","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1002\/(SICI)1522-2616(200001)209:1<83::AID-MANA83>3.0.CO;2-1","article-title":"Atomic and subatomic decompositions in anisotropic function spaces","volume":"209","author":"Farkas","year":"2000","journal-title":"Math. Nachr."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s10444-015-9414-3","article-title":"Vanishing moment conditions for wavelet atoms in higher dimensions","volume":"42","year":"2016","journal-title":"Adv. Comput. Math."},{"key":"ref_64","first-page":"80","article-title":"Wavelet decompositions of anisotropic Besov spaces","volume":"239","author":"Tabacco","year":"2002","journal-title":"Math. Nachr."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1017\/S001309150300107X","article-title":"Wavelet characterizations for anisotropic Besov spaces with 0 < p < 1","volume":"47","author":"Hochmuth","year":"2004","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1006\/acha.2001.0377","article-title":"Wavelet characterizations for anisotropic Besov spaces","volume":"12","author":"Hochmuth","year":"2002","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_67","first-page":"85","article-title":"On the fractional anisotropic Wiener field","volume":"16","author":"Kamont","year":"1996","journal-title":"Probab. Math. Statist."},{"key":"ref_68","unstructured":"Rosiene, C.-P., and Nguyen, T.-Q. (June, January 30). Tensor-product wavelet vs. Mallat decomposition: A comparative analysis. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems VLSI (Cat. No.99CH36349), Orlando, FL, USA."},{"key":"ref_69","unstructured":"Triebel, H. (2006). Theory of Function Spaces III, Birkh\u00e4user. Monographs in Mathematics, 78."},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Triebel, H. (2004). Wavelet Bases in Anisotropic Function Spaces. Funct. Space Differ. Oper. Nonlinear Anal., 370\u2013387.","DOI":"10.4064\/bc64-0-15"},{"key":"ref_71","first-page":"615","article-title":"Wavelet bases in spaces of differentiable functions of anisotropic smoothness. (Russian)","volume":"324","author":"Berkolako","year":"1992","journal-title":"Dokl. Akad. Nauk."},{"key":"ref_72","first-page":"35","article-title":"Unconditional bases in spaces of functions of anisotropic smoothness. (Russian)","volume":"204","author":"Berkolako","year":"1993","journal-title":"Trudy Mat. Inst. Steklov. Issled. Teor. Differ. Funktsii Mnogikh Peremen. Prilozh."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s003659900060","article-title":"Hyperbolic wavelet approximation","volume":"14","author":"DeVore","year":"1998","journal-title":"Constr. Approx."},{"key":"ref_74","unstructured":"Westerink, P.-H. (1989). Subband Coding of Images. [Ph.D. Thesis, Delft University of Technology]."},{"key":"ref_75","first-page":"608619","article-title":"Translation and direction invariant denoising of 2D and 3D images: Experience and algorithms","volume":"2825","author":"Yu","year":"1996","journal-title":"Proc. SPIE"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s10851-007-0777-z","article-title":"Image approximation by rectangular wavelet transform","volume":"27","author":"Zavadsky","year":"2007","journal-title":"J. Math. Imaging Vis."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1109\/MSP.2002.1028352","article-title":"Stochastic Fractal Models for Image Processing","volume":"19","year":"2002","journal-title":"IEEE Signal Proces. Mag."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1023\/A:1015260803576","article-title":"Drap brownien fractionnaire","volume":"17","author":"Ayache","year":"2002","journal-title":"Potential Anal."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1142\/S0219691310003766","article-title":"Shearlet transforms and H\u00f6lder regularities","volume":"8","author":"Lakhonchai","year":"2010","journal-title":"Int. J. Wavelets Multiresolut. Inform. Proc."},{"key":"ref_80","first-page":"275","article-title":"Analysis of H\u00f6lder regularities by wavelet-like transforms with parabolic scaling","volume":"3","author":"Nualtong","year":"2005","journal-title":"Thai J. Math."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1214\/aos\/1018031261","article-title":"Wedgelets: Nearly minimax estimation of edges","volume":"27","author":"Donoho","year":"1999","journal-title":"Ann. Stat."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.acha.2010.08.004","article-title":"Analysis and detection of surface discontinuities using the 3D continuous shearlet transform","volume":"30","author":"Guo","year":"2011","journal-title":"Appl. Comp. Harm. Anal."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"2495","DOI":"10.1098\/rsta.1999.0444","article-title":"Ridgelets: A key to higher-dimensional intermittency?","volume":"357","author":"Donoho","year":"1999","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_84","doi-asserted-by":"crossref","unstructured":"Mallat, S. (2001). Challenges for the 21st century. Applied Mathematics Meets Signal Processing, Proceedings of the the International Conference on Fundamental Sciences: Mathematics and Theoretical Physics (ICFS 2000), Singapore, 13\u201317 March 2000, World Scientific.","DOI":"10.1142\/9789812811264_0006"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1007\/s00041-015-9445-7","article-title":"Resolution of the wavefront set using general continuous wavelet transforms","volume":"22","author":"Fell","year":"2016","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_86","doi-asserted-by":"crossref","unstructured":"Sun, G., Leng, J., and Cattani, C. (2018). A framework for circular multilevel systems in the frequency domain. Symmetry, 10.","DOI":"10.3390\/sym10040101"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"169","DOI":"10.4064\/sm-110-2-169-189","article-title":"Isomorphism of some anisotropic Besov and sequence spaces","volume":"110","author":"Kamont","year":"1994","journal-title":"Studia Math."},{"key":"ref_88","first-page":"1","article-title":"Ondelettes et bases hilbertiennes","volume":"1","author":"Meyer","year":"1986","journal-title":"Rev. Mat. Iberoam."},{"key":"ref_89","unstructured":"Meyer, Y. (1990). Ondelettes et Op\u00e9rateurs, Hermann."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1002\/cpa.3160410705","article-title":"Orthonormal bases of compactly supported wavelets","volume":"41","author":"Daubechies","year":"1988","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_91","unstructured":"Rogers, C.A. (1970). Hausdorff Measures, Cambridge University Press."},{"key":"ref_92","unstructured":"Tao, Q., Vai, M.I., and Xu, Y. (2006). Wavelet Leaders in Multifractal Analysis. Wavelet Analysis and Applications, Birkha\u00fcser Verlag. Applied and Numerical Harmonic Analysis."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.physrep.2005.04.001","article-title":"Anisotropy in turbulent flows and in turbulent transport","volume":"414","author":"Biferalea","year":"2005","journal-title":"Phys. Rep."},{"key":"ref_94","unstructured":"Hinze, J.O. (1975). Turbulence, McGraw-Hill."},{"key":"ref_95","unstructured":"Greenspan, H.P. (1968). The Theory of Rotating Fluids, Cambridge University Press."},{"key":"ref_96","first-page":"240","article-title":"Vortex stretching and anisotropic diffusion in the 3D Navier-Stokes equations","volume":"666","author":"Grujic","year":"2016","journal-title":"Contemp. Math."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1137\/1036004","article-title":"Geometric statistics in turbulence","volume":"36","author":"Constantin","year":"1994","journal-title":"SIAM Rev."},{"key":"ref_98","unstructured":"Ran, Z. (2010, December 23). Statistical Theory of Isotropic Turbulence. Part IV: Multiscales and Cascade. Available online: https:\/\/arxiv.org\/pdf\/1012.5151."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/825\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:00:29Z","timestamp":1760187629000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/825"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,21]]},"references-count":98,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["sym11060825"],"URL":"https:\/\/doi.org\/10.3390\/sym11060825","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,6,21]]}}}