[...]... v Ferrata 1, Pavia, Italy e-mail: silvia.bertoluzza@imati.cnr.it S Bertoluzza et al., Multiscale and Adaptivity: Modeling, Numerics and Applications, Lecture Notes in Mathematics 2040, DOI 10.1007/978-3-642-24079-9 1, © Springer-Verlag Berlin Heidelberg 2012 1 2 S Bertoluzza solutions which present singularities, and the ability to represent such solution with as little as possible degrees of freedom... that both ' and have arbitrarily high regularity R, are supported in 0; L/ and they generate by translations and dilations orthonormal bases for the spaces Vj and Wj The projectors Pj are, as in the Haar case, L2 orthogonal projectors Also in this case the scaling function and the dual function Q Q coincide and we have Vj D Vj , Wj D Wj , ' D ' and Q D (see Fig 1 for an Q example of scaling and wavelet... justification and analysis of convergence and computational complexity is available 1 Introduction Wavelet bases were introduced in the late 1980s as a tool for signal and image processing Among the applications considered at the beginning we recall applications in the analysis of seismic signals, the numerous applications in image processing – image compression, edge-detection, denoising, applications. .. Pj W L2 R/ ! Vj and Pj D Pj W 2 Qj , both verifying a commutativity property of the form (4) L R/ ! V (c) Two dual refinable functions ' and ' (the scaling functions) which, by Q Q translation and dilation generate biorthogonal bases for the Vj ’s and the Vj ’s, Qj in the form respectively, and that allow to write the two projectors Pj and P (8) (d) A sequence of complement spaces Wj (and it is easy... and Q 2 V1 and introduce a pair of dual wavelets x/ D X gk '.2x k2Z k; Q x/ D k/ X k gk '.2x Q Q k/: (12) k The following theorem holds [18]: Theorem 1 The integer translates of the wavelet functions and Q are orthogonal to ' and ', respectively, and they form a couple of biorthogonal sequences Q More precisely, they satisfy h ; Q k/i D ı0;k h k/; 'i D h Q Q The projection operator Qj can be expanded... classical construction of wavelet bases for L2 R/ [39], all basis functions ' ; 2 Kj and , 2 rj with j 0, as well as their duals ' and Q , are constructed by translation and dilation of a single scaling Q function ' and a single mother wavelet (resp ' and Q ) Clearly, the properties Q of the function will transfer to the functions and will imply properties of the corresponding wavelet basis To start with, we... what is done for Daubechies’ wavelets, it is possible [18] to characterize and construct a family of sequences Q hk /k for which the solution to the refinement equation (9) exists, is dual to the Q B-spline BN , has compact support and arbitrarily high smoothness R Figures 2 Q Q and 3 show the functions ', ', and Q for N D 1, R D 0 and N D 1, R D 1, Q respectively Adaptive Wavelet Methods 13 rbio2.2 :... that there exist an integer M > 0 and an integer R > 0, with M > R, such that for n D 0; : : : ; M and for s such that 0 Ä s Ä R one has dn O 0/ D 0 d n (a) Z and 1 C j j2 /s j O /j2 d < 1: (b) (30) R Q Q Analogously, for Q we assume that there exist integers M > R > 0 such that for Q and for s such that 0 Ä s Ä R one has Q n D 0; : : : ; M Q dn O 0/ D 0 n d (a) Z and Q 1 C j j2 /s j O /j2 d < 1:... inclusion Vj the contrary, property (4) is not verified by general non-orthogonal projectors and expresses the fact that the approximation Pj f can be derived from Pj C1 f without any further information on f Equations (5) and (6) require that the projector Pj respects the translation and dilation invariance properties (iii) and (iv) of the MRA Since f'0;k g is a Riesz basis for V0 there exists a biorthogonal... : psi dec 1.5 2 1.5 1 1 0.5 0.5 0 −0.5 0 0 1 2 3 4 5 −1 0 1 rbio2.2 : phi rec 2 3 4 5 4 5 rbio2.2 : psi rec 6 10 4 5 2 0 0 −2 −4 0 1 2 3 4 5 −5 0 1 2 3 Fig 2 Scaling and wavelet functions ' and for decomposition (top) and the duals ' and Q for Q reconstruction (bottom) corresponding to the biorthogonal basis B2.2 2.3.3 Interpolating Wavelets It is also interesting to consider an example where Lj is . Nochetto Alfio Quarteroni  Kunibert G. Siebert Andreas Veeser Multiscale and Adaptivity: Modeling, Numerics and Applications C.I.M.E. Summer School, Cetraro,. CIME-EMS Summer School in applied mathematics on Multiscale and Adaptivity: Modeling, Numerics and Applications was held in Cetraro (Italy) from July