Nội suy trên lưới thưa và sản phẩm tensor của không gian Nikol'skij–Besov

Winfried Sickel1, Frauke Sprengel2
1Mathematisches Institut, F.-Schiller-Universität, Jena, Germany
2Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands

Tóm tắt

Chúng tôi nghiên cứu thứ tự hội tụ của nội suy định kỳ trên các lưới thưa (nội suy trộn) trong khuôn khổ sản phẩm tensor của không gian Nikol'skij–Besov. Để đạt được điều này, chúng tôi sử dụng tính đồng nhất của các chuẩn tensor được xem xét và cung cấp một phương pháp thống nhất để ước lượng sai số cho việc nội suy các hàm định kỳ đơn biến từ các không gian Nikol'skij–Besov.

Từ khóa

#nội suy định kỳ #lưới thưa #không gian Nikol'skij–Besov #sai số nội suy #sản phẩm tensor

Tài liệu tham khảo

T. I. Amanov, Spaces of Differentiable Functions with Dominating Mixed Derivatives, Nauka Kaz. SSR, Alma-Ata (1976). V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Preprint, University of Erlangen (1998). G. Baszenski and F.-J. Delvos, A discrete Fourier transform scheme for Boolean sums of trigonometric operators, in Multivariate Approximation Theory IV (C. K. Chui, W. Schempp, and K. Zeller, eds.), ISNM 90, Birkhäuser, Basel (1989), pp. 35-46. G. Brumme, Error estimates for periodic interpolation by translates, in Wavelets, Images, and Surface Fitting (P. J. Laurent, A. L. Méhauté, and L. L. Schumaker, eds.), AK Peters, Boston, (1994), pp. 75-82. F.-J. Delvos, Trigonometric approximation in multivariate periodic Hilbert spaces, in Multivariate Approximation, Recent Trends and Results (W. Haussmann, K. Jetter, and M. Reimer, eds.), Akademie-Verlag, Berlin, (1997), pp. 35-44. F.-J. Delvos and W. Schempp, Boolean Methods in Interpolation and Approximation, Pitman Research Notes in Mathematics Series 230, Longman Scientific and Technical, Harlow, (1989). R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov, Hyperbolic wavelet approximation, Constructive Approximation 14, 1-26, (1998). R. A. DeVore, P. P. Petrushev, and V. N. Temlyakov, Multivariate trigonometric polynomial approximation with frequencies from the hyperbolic cross, Mat. Zametki 56, 36-63 (1994) (English Transl. in Math. Notes, 56, 1994). R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin (1993). B. Jawerth, Some observations on Besov and Triebel-Lizorkin spaces, Math. Scand. 40, 94-104 (1977). K. Jetter, Multivariate approximation: A view from cardinal interpolation, in Approximation Theory VII (E. W. Cheney, C. K. Chui, and L. L. Schumaker, eds.), Academic Press, New York, (1992), pp. 131-167. R.-Q. Jia and J. Lei, Approximation by multiinteger translates of functions having global support, J. Approx. Theory 72, 2-23 (1993). W. A. Light and E. W. Cheney, Approximation Theory in Tensor Product Spaces, Lecture Notes in Math. 1169, Springer, Berlin (1985). G. G. Lorentz, M. von Golitschek, and Y. Makovoz, Constructive Approximation: Advanced Problems, Springer, Berlin (1996). S. M. Nikol'skij, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin (1975). P. Oswald, On estimates for hierarchic basis representations of finite element functions, Forschungsergebnisse FSU Jena, N/89/16, (1989). A. Pietsch, Approximation spaces, J. Approx. Theory 32, 115-134 (1981). G. Pöplau and F. Sprengel, Some error estimates for periodic interpolation on full and sparse grids, in Curves and Surfaces with Applications in CAGD (A. L. Méhauté, C. Rabut, and L. L. Schumaker, eds.), Vanderbilt University Press, Nashville, TN (1997), pp. 355-362. J. Prestin and K. Selig, Interpolatory and orthonormal trigonometric wavelets, in Signal and Image Representation in Combined Spaces (J. Zeevi and R. Coifman, eds.), Academic Press, New York, (1998), pp. 201-255. S. Ries and R. L. Stens, Approximation by generalized sampling series, in Constr. Theory of Functions '84, Publ. House of the Bulgarian Academy of Science, Sofia (1984), pp. 746-756. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, Berlin (1996). H.-J. Schmeisser, An unconditional basis in periodic spaces with dominating mixed smoothness properties, Anal. Math. 13, 153-168 (1987). H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley, Chichester (1987). W. Sickel, Tensor products of Besov spaces, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/99/04 (1999). F. Sprengel, Interpolation and Wavelet Decomposition of Multivariate Periodic Functions, PhD thesis, University of Rostock (1997) (German). F. Sprengel, A tool for approximation in bivariate periodic Sobolev spaces, in Approximation Theory IX (C. K. Chui and L. L. Schumaker, eds.), Vol. 2, Vanderbilt University Press, Nashville, TN, (1998), pp. 319-326. V. N. Temlyakov, Approximation of Periodic Functions, Nova Science, New York (1993). V. M. Tichomirov, Approximation theory, in Encyclopedia of Mathematical Science: Analysis II (R. V. Gamkrelidze, ed.), Vol. 14, Springer, Berlin (1990), pp. 93-244. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Deutscher Verlag der Wissenschaften, Berlin (1978). H. Triebel, Theory of Function Spaces, Birkhäuser, Basel (1983).