Abstract
An orthonormal wavelet basis on the circle γ is constructed. By estabishing some new theorems on complex spline functions, the \(\mathop L\limits^ \circ _2 (I)\) space can be decomposed into different orthogonal subspaces. Formulas of decomposition and reconstruction imply only two terms.
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References
H.L. Chen, Quasiinterpolant splines on the unit circle. J. Approx. Theory Vol. 38, No. 4 (1983), 312–318.
H.L. Chen, Interpolation by splines on finite and infinite planar sets. Chin. Ann. of Math. 5B(3) (1984), 375–390.
H.L. Chen, Tron Hvaring, Approximation of complex harmonic functions by complex harmonic splines. Math. Comp. 42 (1984), 151–164.
H.L. Chen, Interpolation and approximation on the unit disc by complex harmonic splines. J. Approx. Theory Vol. 43, No. 2 (1985), 112–123.
G. Walz, Spline Funktionen im Komplexen. BI-Wiss-Verl. Mannheim; Wien; Zurich (1991).
Y. Meyer, Wavelets and Operators,Cambridge Univ. Press, Cambridge 1993.
I. Daubechies, Ten Lectures on Wavelets, CBMS, 61, SIAM, Philadelphia, 1992.
V. Perrier, C. Basdevant, Periodical wavelet analysis, a tool for inhomogeneous field investigation. Theory and algorithms, Rech. Aérospat. 1989-3, 53-67.
G. Plonka, M. Tasche, A unified approach to periodic wavelets, in: Wavelets: Theory, Algorithms, and Applications, C. K. Chui, L. Montefusco, and L. Puccio (eds.), (1994), 137-151.
Y. W. Koh, S.L. Lee, H. H. Tan, Periodic orthogonal splines and wavelets, in Appl. Comput. Harmonic Anal., 2(1995), 201–218.
S. Dahlke, Multiresolution analysis and wavelets on locally compact abelian groups. in Wavelets, Images and Surface Fitting, P.L. Laurent, A. Le Méhauté and L. L. Schumaber (eds), AK Peters 1994, 141-156.
M. Kamada, K. Toraichi and R. Mori, Periodic orthonormal Bases, J. Appr. Th. 55, 27–34(1988).
H.L. Chen, Construction of orthonormal wavelet basis in periodic case. to appear in Chinese Science Bulletin.
H.L. Chen, Wavelets from trigonometric spline approach, to appear in Journal of Approximation Theory and its Applications.
I.J. Schoenberg, On polynomial spline functions on the circle I, II. In: Proc. Conf. Constructive Theory of Functions, G. Alexits and S.B. Steckin (eds.), Budapest (1971), 403-433.
C.de. Boor and G.J. Fix, spline approximation by quasiinterpolants, J. Approx. Theory 8 (1973), 19–45.
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Chen, HL. Spline Wavelets on the Unit Circle. Results. Math. 31, 322–336 (1997). https://doi.org/10.1007/BF03322168
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DOI: https://doi.org/10.1007/BF03322168