List with Selected Abstracts

A dilation equation is a functional equation of the form f(t) = ∑N k=0 ck f(2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function in terms of the joint spectral radius of two matrices determined by the coefficients {c0, . . . , cN}. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions. 11. C. Heil, Methods of solving dilation equations, in: “Probabilistic and Stochastic Methods in Analysis, with Applications” (Il Ciocco, 1991), J. S. Byrnes, et al., eds., NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 372, Kluwer Acad. Pub., Dordrecht (1992), pp. 15–45. MR 93h:42023. Zbl. 0761.93040. 12. J. Benedetto, C. Heil, and D. Walnut, Uncertainty principles for time-frequency operators, in: “Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations,” Oper. Theory Adv. Appl., 58, I. Gohberg, ed., Birkhäuser, Basel (1992), pp. 1–25. MR 94a:42041. Zbl. 0790.42017. 13. C. Heil and D. Colella, Dilation equations and the smoothness of compactly supported wavelets, in: “Wavelets: Mathematics and Applications,” J. J. Benedetto and M. W. Frazier, eds., CRC Press, Boca Raton, FL (1994), pp. 163–201. MR 94j:42049. Zbl. 0882.42025. Abstract. The construction of compactly supported wavelets with specified amounts of smoothness is an important problem in wavelet theory. This problem reduces to the construction of scaling functions, i.e., solutions f of dilation equations f(t) = ∑N k=0 ck f(2t − k), with specified smoothness. This article characterizes all smooth, compactly supported scaling functions in terms of a joint spectral radius of two N ×N matrices T0, T1 constructed from the coefficients {c0, . . . , cN} of the dilation equation, restricted to an appropriate subspace of C . The number of continuous derivatives of the scaling function and the range of Hölder exponents of continuity of the last continuous derivative are determined by the value of this joint spectral radius. Numerous examples are provided to illustrate the results. The construction of compactly supported wavelets with specified amounts of smoothness is an important problem in wavelet theory. This problem reduces to the construction of scaling functions, i.e., solutions f of dilation equations f(t) = ∑N k=0 ck f(2t − k), with specified smoothness. This article characterizes all smooth, compactly supported scaling functions in terms of a joint spectral radius of two N ×N matrices T0, T1 constructed from the coefficients {c0, . . . , cN} of the dilation equation, restricted to an appropriate subspace of C . The number of continuous derivatives of the scaling function and the range of Hölder exponents of continuity of the last continuous derivative are determined by the value of this joint spectral radius. Numerous examples are provided to illustrate the results.