caa a22 4500 606204954 CHVBK 20210128101005.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00034-014-9937-8 doi (NATIONALLICENCE)springer-10.1007/s00034-014-9937-8 On Relating Interpolatory Wavelets to Interpolatory Scaling Functions in Multiresolution Analyses [Elektronische Daten] [Zhiguo Zhang, Mark Kon] Construction of interpolatory wavelets is an important topic in discrete signal processing. In classical wavelet sampling theories, interpolatory wavelets are constructed from Riesz bases in wavelet spaces. Since analytical expressions for such Riesz bases are generally complex or unavailable, it has been difficult to obtain suitable interpolatory wavelets in practice. In this paper, interpolatory scaling functions are used to determine and construct interpolatory wavelets. We first show that there may not exist interpolatory wavelets even when interpolatory scaling functions exist. Then, an inequality in terms of interpolatory scaling functions, denoted as the two-scale condition, is given as a necessary and sufficient condition for existence of interpolatory wavelets. Finally, based on the two-scale condition, a filter bank is constructed for obtaining interpolatory wavelets directly from interpolatory scaling functions. In examples, our theorems are applied to some typical wavelet spaces, demonstrating our construction algorithm for interpolatory wavelets. Springer Science+Business Media New York, 2014 Wavelet sampling nationallicence Interpolatory basis nationallicence Interpolatory wavelet nationallicence Multiresolution analysis nationallicence Filter bank nationallicence $$W_j :$$ W j : : Wavelet space nationallicence $$V_j :$$ V j : : Approximation space of MRA nationallicence $$L^2(\mathbb {R}):$$ L 2 ( R ) : : Square integrable function nationallicence $$l^2(\mathbb {R}):$$ l 2 ( R ) : : Finite energy discrete signals $$\sum \nolimits _{k=-\infty }^{+\infty } {\left| {f_s (k)} \right| ^2} <+\infty $$ ∑ k = - ∞ + ∞ f s ( k ) 2 < + ∞ nationallicence $$f_s (x):$$ f s ( x ) : : Signal to be recovered nationallicence $$f_{ap} (x):$$ f a p ( x ) : : Recovery of signal nationallicence $$\hat{f}(w):$$ f ^ ( w ) : : Fourier transform of $$f(x)$$ f ( x ) nationallicence $$\bar{f}(x):$$ f ¯ ( x ) : : Complex conjugate of $$f(x)$$ f ( x ) nationallicence $$\phi (x):$$ ϕ ( x ) : : Scaling function of $$V_0 $$ V 0 nationallicence $$\psi (x):$$ ψ ( x ) : : Wavelet of $$W_0 $$ W 0 nationallicence $$\{S^\phi (2^jx-k)\}_{k\in \mathbb {Z}} :$$ { S ϕ ( 2 j x - k ) } k ∈ Z : : Interpolatory basis relative to the samples $$\{f_s (k/2^j)\}_{k\in \mathbb {Z}} $$ { f s ( k / 2 j ) } k ∈ Z for $$V_j $$ V j nationallicence $$\{\tilde{S}^\phi (2^jx-k)\}_{k\in \mathbb {Z}} :$$ { S ~ ϕ ( 2 j x - k ) } k ∈ Z : : Dual basis of $$\{S^\phi (2^jx-k)\}_{k\in \mathbb {Z}} $$ { S ϕ ( 2 j x - k ) } k ∈ Z for $$V_j $$ V j nationallicence $$\{S^\psi (2^jx-k)\}_{k\in \mathbb {Z}}:$$ { S ψ ( 2 j x - k ) } k ∈ Z : : Interpolatory basis relative to the samples $$\{f_s (k/2^j+1/2^{j+1})\}_{k\in \mathbb {Z}} $$ { f s ( k / 2 j + 1 / 2 j + 1 ) } k ∈ Z for $$W_j $$ W j nationallicence $$\{\tilde{S}^\psi (2^jx-k)\}_{k\in \mathbb {Z}}:$$ { S ~ ψ ( 2 j x - k ) } k ∈ Z : : Dual basis of $$\{S^\psi (2^jx-k)\}_{k\in \mathbb {Z}} $$ { S ψ ( 2 j x - k ) } k ∈ Z for $$W_j $$ W j nationallicence $$P_\phi (w):$$ P ϕ ( w ) : : $$l^2$$ l 2 -Sequence defined in (41) nationallicence $$P_s (w)_{:}$$ P s ( w ) : : $$l^2$$ l 2 -Sequence defined in (3) nationallicence $$Q_\psi (w):$$ Q ψ ( w ) : : $$l^2$$ l 2 -Sequence defined in (18) nationallicence $$Q_s (w):$$ Q s ( w ) : : $$l^2$$ l 2 -Sequence defined in (14) nationallicence $$E_s (w):$$ E s ( w ) : : Period function defined in (7) nationallicence $$E_\phi (w):$$ E ϕ ( w ) : : Period function defined in (20) nationallicence $$\Delta _{P_s ,Q_s } :$$ Δ P s , Q s : : Function defined in (16) nationallicence $$T_{W_{j-1} }^{1/2} :$$ T W j - 1 1 / 2 : : Sampling operator defined in (5) nationallicence Zhang Zhiguo College of Automation, University of Electronic Science and Technology of China, 610000, Chengdu, China aut Kon Mark Department of Mathematics and Statistics, Boston University, 02215, Boston, MA, USA aut Circuits, Systems, and Signal Processing Springer US; http://www.springer-ny.com 34/6(2015-06-01), 1947-1976 0278-081X 34:6<1947 2015 34 34 https://doi.org/10.1007/s00034-014-9937-8 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00034-014-9937-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Zhang Zhiguo College of Automation, University of Electronic Science and Technology of China, 610000, Chengdu, China aut NATIONALLICENCE 700 1- Kon Mark Department of Mathematics and Statistics, Boston University, 02215, Boston, MA, USA aut NATIONALLICENCE 773 0- Circuits, Systems, and Signal Processing Springer US; http://www.springer-ny.com 34/6(2015-06-01), 1947-1976 0278-081X 34:6<1947 2015 34 34