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Math. Methods Appl. Sci. 39, 2685 (2015)\nAcar, T., Aral, A., Mohiuddine, S.A.: Approximation by bivariate \\((p, q)\\)-Bernstein–Kantorovich operators. Iran. J. Sci. Technol. Trans. A Sci (2016). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs40995-016-0045-4\nAcar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modification of \\((p, q)\\)-Baskakov operators. J. Inequal. Appl. 98, 98 (2016)\nBukweli-Kyemba, J.D., Hounkonnou, M.N.: Quantum deformed algebras: coherent states and special functions (2013). arXiv:1301.0116v1\nBurban, I.M., Klimyk, A.U.: \\((P, Q)\\)-Differentiation, \\((P, Q)\\) integration, and \\((P, Q)\\)-hypergeometric functions related to quantum groups. Integral Transform Spec. Funct. 2(1), 15–36 (1994)\nChakrabarti, R., Jagannathan, R.: A \\((p, q)\\)-oscillator realization of two-parameter quantum algebras. J. Phys. A Math. 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Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713–747.\nC. A. Micchelli and Yuesheng Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comp. Harmonic Anal. 1 (1994), 391–401.\nC. A. Micchelli and Yuesheng Xu, Reconstruction and decomposition algorithms for biorthogonal multiwavelets, Multidimensional Systems and Signal Processing 8 (1997), 31–69.\nC. A. Micchelli, Yuesheng Xu and Y. Zhao, Wavelet Galerkin methods for second kind integral equations, J. Comp. Appl. Math. 86 (1997), 251–270.\nT. Sauer and Yuan Xu, On multivariate Lagrange interpolation, Math. Comp. 64 (1995), 1147–1170.\nT. Sauer and Yuan Xu, Regular points for Lagrange interpolation on the unit disk, Numer. Algo. 12 (1996), 287–296.\nH. Yserentant, On the multilevel splitting of finite element spaces, Numer. 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L.V.: Complex Analysis, 3rd edn. International Series in Pure and Applied Mathematics, p. 331. McGraw-Hill Book Co., New York, (1978). An introduction to the theory of analytic functions of one complex variable\nAkamine, S., Fujino, H.: Reflection principle for lightlike line segments on maximal surfaces. Ann. Global Anal. Geom. 59(1), 93–108 (2021). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs10455-020-09743-4\nAkamine, S., Fujino, H.: Duality of boundary value problems for minimal and maximal surfaces. To appear in Comm. Anal. Geom. arXiv:1909.00975\nAkamine, S., Fujino, H.: Extension of Krust theorem and deformations of minimal surfaces. Ann. Mat. Pura Appl. (4) https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs10231-022-01211-z.\nda Silva, L.C.B.: Holomorphic representation of minimal surfaces in simply isotropic space. J. Geom. 112(3), 35–21 (2021). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs00022-021-00598-z\nDierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces, 2nd edn. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 339, p. 688. Springer. With assistance and contributions by A. Küster and R. Jakob. (2010). https:\u002F\u002Fdoi.org\u002F10.1007\u002F978-3-642-11698-8\nKatznelson, Y.: An Introduction to Harmonic Analysis. In: Cambridge Mathematical Library, 3rd edn., p. 314. Cambridge University Press, Cambridge (2004). https:\u002F\u002Fdoi.org\u002F10.1017\u002FCBO9781139165372\nMa, X., Wang, C., Wang, P.: Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space. Adv. Math. 249, 311–347 (2013). https:\u002F\u002Fdoi.org\u002F10.1016\u002Fj.aim.2013.09.013\nMilnor, J.: Dynamics in One Complex Variable. In: Annals of Mathematics Studies, vol. 160, 3rd edn., p. 304. Princeton University Press, Princeton, NJ (2006)\nPember, M.: Weierstrass-type representations. Geom. Dedicata. 204, 299–309 (2020). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs10711-019-00456-y\nPottmann, H., Grohs, P., Mitra, N.J.: Laguerre minimal surfaces, isotropic geometry and linear elasticity. Adv. Comput. Math. 31(4), 391–419 (2009). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs10444-008-9076-5\nSachs, H.: Isotrope Geometrie des Raumes, p. 323. Friedr. Vieweg & Sohn, Braunschweig (1990). https:\u002F\u002Fdoi.org\u002F10.1007\u002F978-3-322-83785-1\nSato, Y.: \\(d\\)-minimal surfaces in three-dimensional singular semi-Euclidean space \\({\\mathbb{R} }^{0,2,1}\\). Tamkang J. Math. 52(1), 37–67 (2021). https:\u002F\u002Fdoi.org\u002F10.5556\u002Fj.tkjm.52.2021.3045\nSeo, J.J., Yang, S.-D.: Zero mean curvature surfaces in isotropic three-space. Bull. Korean Math. Soc. 58(1), 1–20 (2021). https:\u002F\u002Fdoi.org\u002F10.4134\u002FBKMS.b190783\nStrubecker, K.: Differentialgeometrie des isotropen Raumes. III. Flächentheorie. Math. Z. 48, 369–427 (1942). https:\u002F\u002Fdoi.org\u002F10.1007\u002FBF01180022\nStrubecker, K.: Über Potentialflächen. Arch. Math. 5, 32–38 (1954). https:\u002F\u002Fdoi.org\u002F10.1007\u002FBF01899315\nStrubecker, K.: Duale Minimalflächen des isotropen Raumes. Rad Jugoslav. Akad. Znan. Umjet. 382, 91–107 (1978)",{"EN":1037},"Zero mean curvature surfaces in the simply isotropic 3-space \n                \n                  \n                \n                $${\\mathbb {I}}^3$$\n                \n               naturally appear as intermediate geometry between geometry of minimal surfaces in \n                \n                  \n                \n                $${\\mathbb {E}}^3$$\n                \n               and that of maximal surfaces in \n                \n                  \n                \n                $${\\mathbb {L}}^3$$\n                \n              . In this paper, we investigate reflection principles for zero mean curvature surfaces in \n                \n                  \n                \n                $${\\mathbb {I}}^3$$\n                \n               as with the above surfaces in \n                \n                  \n                \n                $${\\mathbb {E}}^3$$\n                \n               and \n                \n                  \n                \n                $${\\mathbb {L}}^3$$\n                \n              . In particular, we show a reflection principle for isotropic line segments on such zero mean curvature surfaces in \n                \n                  \n                \n                $${\\mathbb {I}}^3$$\n                \n              , along which the induced metrics become singular.",{"EN":1039},"Reflection Principles for Zero Mean Curvature Surfaces in the Simply Isotropic 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Allyn and Bacon, Boston (1966)\nGalego, E.M., da Silva, A.L.P.: An optimal nonlinear extension of Banach–Stone theorem. J. Funct. Anal. 271, 2166–2176 (2016)\nGarrido, M.I., Jaramillo, J.A.: Variations on the Banach–Stone theorem. Extracta Math. 17, 351–383 (2002)\nHyers, D.H., Ulam, S.M.: On approximate isometries. Bull. Am. Math. Soc. 51, 288–292 (1945)\nMazur, S., Ulam, S.: Sur les transformations isométriques d’espaces vectoriels normés. C. R. Acad. Sci. Paris 194, 946–948 (1932)\nOmladič, M., Šemrl, P.: On non linear perturbations of isometries. Math. Ann. 303, 617–628 (1995)\nPhelps, R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)\nŠemrl, P., Väisälä, J.: Nonsurjective nearisometries of Banach spaces. J. Funct. Anal. 198, 268–278 (2003)\nStone, M.H.: Applications of the theory of boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)\nSun, L.: Hyers–Ulam stability of \\(\\varepsilon \\)-isometries between the positive cones of \\(L^p\\)-spaces. J. Math. Anal. Appl. (2020). https:\u002F\u002Fdoi.org\u002F10.1016\u002Fj.jmaa.2020.124014\nSun, L.: A note on stability of non-surjective \\(\\varepsilon \\)-isometries between the positive cones of \\(L^p\\)-spaces. Indian J. Pure Appl. Math. (2021). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs13226-021-00047-2\nVäisälä, J.: Isometric approximation property in Euclidean spaces. Isr. J. Math. 128, 1–27 (2002)\nVäisälä, J.: Isometric approximation property of unbounded sets. Result Math. 43, 359–372 (2003)\nVestfrid, I.A.: \\(\\varepsilon \\)-isometries in Euclidean spaces. Nonlinear Anal. 63, 1191–1198 (2005)\nVestfrid, I.A.: \\(\\varepsilon \\)-isometries in \\(\\ell _\\infty ^n\\). Nonlinear Funct. Anal. Appl. 12, 433–438 (2007)\nVestfrid, I.A.: Near-isometries on unit sphere. Ukr. Math. J. 72, 663–670 (2020)\nZhou, Y., Zhang, Z., Liu, C.: On linear isometries and \\(\\varepsilon \\)-isometries between Banach spaces. J. Math. Anal. App. 435, 754–764 (2016)\nZhou, Y., Zhang, Z., Liu, C.: On representation of isometric embeddings between Hausdorff metric spaces of compact convex subsets. Houston J. Math. 44, 917–925 (2018)",{"EN":1231},"Let X, Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C(X) be the real Banach space of all continuous functions on X, and \n                \n                  \n                \n                $$C_+(X)$$\n                \n               be the positive cone of C(X). In this paper, we show that if there exists a \n                \n                  \n                \n                $$\\delta $$\n                \n              -surjective \n                \n                  \n                \n                $$\\varepsilon $$\n                \n              -isometry \n                \n                  \n                \n                $$F: C_+(X)\\rightarrow C_+(Y)$$\n                \n              , then X and Y are homeomorphic. 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