Approximation properties of interpolation and quasi-interpolation operators

• Applicant: Dr. Yurii Kolomoitsev
• Subject Area: Mathematics
• Term: from 2018 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 406704922

• Final Report Year: 2022

Final Report Abstract

Interpolation and quasi-interpolation are among the most important mathematical methods used in many branches of science and engineering. They play a crucial role as a connecting link between continuous-time and discrete-time signals. Because of the trend to replace analog signals by digital ones, interpolation and quasi-interpolation have found applications in many fields including signal analysis, information theory, image processing, acoustics, optics, medical imaging, etc. For proper application of interpolation and quasi-interpolation operators, it is very important to know the quality of approximation of functions by such operators in various settings. In this project, we studied approximation properties of wide classes of interpolation and quasi-interpolation (quasi-projection) operators in the classical function spaces on the multivariate Euclidean space, torus, and hypercube. In particular, we obtained a series of new error estimates for interpolation and quasi-interpolation operators by developing a unified approach based on the theory of Fourier multipliers. In some important cases, we showed that our estimates of the approximation error are sharp in the sense that they are equivalent to an appropriate modulus of smoothness or K-functional. One of the main features of our results is that they can be used in applied problems, which contain noisy data and the functional information is provided by other means than point evaluation (for example integrals, averaging, divided differences etc) or in the case when information is provided by point evaluation but corresponds to discontinuous, unbounded, or highly oscillating signals. The results can be also used in applied high-dimensional problems, which naturally appear in various scientific areas. To achieve the goals of the project, we investigated also various measures of smoothness that depending on the tasks considered (type of the operator and the function space) provide full and adequate information about the quality of approximation of a given function by the corresponding operator. In particular, we studied properties of such objects of harmonic analysis and approximation theory as the Lebesgue constants of interpolation processes and partial sums of Fourier series and different measures of smoothness (special moduli of smoothness, K-functionals, and their realizations and the corresponding semi-discrete quantities). In our research, we paid special attention to the anisotropic nature of the studied objects.

Publications

• Approximation by periodic multivariate quasi-projection operators. Journal of Mathematical Analysis and Applications, 489(2), 124192.
  Kolomoitsev, Yu.; Krivoshein, A. & Skopina, M.
  
• Properties of moduli of smoothness in Lp(Rd). Journal of Approximation Theory, 257, 105423.
  Kolomoitsev, Yurii & Tikhonov, Sergey
  
• Smoothness of functions versus smoothness of approximation processes. Bulletin of Mathematical Sciences, 10(03), 2030002.
  Kolomoitsev, Yu. S. & Tikhonov, S. Yu.
  
• Approximation by multivariate quasi-projection operators and Fourier multipliers. Applied Mathematics and Computation, 400, 125955.
  Kolomoitsev, Yurii & Skopina, Maria
  
• Approximation properties of periodic multivariate quasi-interpolation operators. Journal of Approximation Theory, 270, 105631.
  Kolomoitsev, Yurii & Prestin, Jürgen
  
• Asymptotics of the Lebesgue constants for a $d$-dimensional simplex. Proceedings of the American Mathematical Society, 149(7), 2911-2926.
  Kolomoitsev, Yurii & Liflyand, Elijah
  
• Asymptotics of the Lebesgue constants for bivariate approximation processes. Applied Mathematics and Computation, 403, 126192.
  Kolomoitsev, Yurii & Lomako, Tetiana
  
• Quasi-projection operators in weighted L spaces. Applied and Computational Harmonic Analysis, 52, 165-197.
  Kolomoitsev, Yu. & Skopina, M.
  
• Approximation by quasi-interpolation operators and Smolyak's algorithm. Journal of Complexity, 69, 101601.
  Kolomoitsev, Yurii
  
• Uniform approximation by multivariate quasi-projection operators. Analysis and Mathematical Physics, 12(2).
  Kolomoitsev, Yu. & Skopina, M.