Interpolatory And Orthonormal Trigonometric. Wavelets. Jurgen Prestin. and Kathi Selig

Transcription

1 Interolatory And Orthonormal Trigonometric Wavelets Jurgen Prestin and Kathi Selig Abstract. The aim of this aer is the detailed investigation of trigonometric olynomial saces as a tool for aroximation and signal analysis. Samle saces are generated by equidistant translates of certain de la Vallee Poussin means. The dierent de la Vallee Poussin means enable us to choose between better time- or frequency-localization. For nested samle saces and corresonding wavelet saces, we discuss dierent bases and their transformations. x Introduction Trigonometric olynomials and the aroximation of eriodic functions by olynomials lay an imortant role in harmonic analysis. Here we are interested in constructing time-localized bases for certain saces of trigonometric olynomials. We use de la Vallee Poussin means of the usual Dirichlet kernel, which allow the investigation of simle roections onto these saces. Interolation and orthogonal roection are discussed. With the dierent de la Vallee Poussin means, the oerator norms of these roections as well as the time-frequency localization of the basis functions can be controlled. In order to imrove the time-localization, the side oscillations are reduced by including more frequencies and averaging the highest ones. Adating basic ideas of wavelet theory, interolatory and orthonormal bases are emloyed, both of which are constructed from equidistant shifts of a single olynomial. One of our main goals is the investigation of basis transformations in a form that facilitates fast algorithms. We observe that the corresonding transformation matrices have a circulant structure and can be diagonalized by Fourier matrices. The resulting diagonal matrices contain the eigenvalues, which are comuted exlicitly. So, the algorithms can be easily realized using the Fast Fourier Transform (FFT). Further, we consider the nesting of the samle saces to obtain multiresolution analyses (RA's). For the resulting orthogonal wavelet saces, we roceed as above and nd wavelet bases consisting of translates of a single olynomial. Again, interolatory and orthonormal bases are constructed, which show the same time-frequency behaviour as the samle bases do. The basis transformations can be described analogously by circulant matrices. ost imortant for ractical reasons are the decomosition of signals in frequency bands, which corresond to the wavelet saces, and their reconstruction. Signal and Image Reresentation in Combined Saces Josh Zeevi and Ronald Coifman (Eds.),. {54. Coyright c995 by Academic Press, Inc. All rights of reroduction in any form reserved. ISB --xxxxxx-x

2 J. Prestin & K. Selig Focusing on basis transformations, the two-scale relations and decomosition formulas are also given in matrix notation suitable for the use of FFT methods. Then the transformations for signal data follow easily (see [9]). This direct aroach to trigonometric wavelets is an alternative to the construction of eriodic wavelets by eriodization of cardinal wavelets (see [], [3], [6], [8], []). In articular, the de la Vallee Poussin means can be obtained by eriodization of cardinal functions which are roducts of sinus cardinalis functions. A rst constructive aroach to trigonometric wavelets was introduced by C. K. Chui and H.. haskar in [] taking very simle nested multiresolution saces and corresonding wavelet saces. They constructed the scaling functions by means of the artial Fourier sum oerator alied to a Haar function with small suort in order to get time-localized basis functions. The two disadvantages of this rocedure are, rst, that the resulting aroximation in the samle sace is only a quasi-interolation and, second, that the basis functions are ust as localized as the simlest de la Vallee Poussin mean, the modied Dirichlet kernel. The idea that we followed was initiated by the aer [8] of A. A. Privalov, where he considered the de la Vallee Poussin means in a dierent content. amely, he was interested in an orthonormal olynomial Schauder basis for C of otimal degree. This roblem has also been considered by P. Wotaszczyk and K. Wozniakowski in [3], by D. On and K. Oskolkov in [9], and was nally solved by R. Lorentz and A. A. Sahakian in [7] (see also K. Wozniakowski [4]). Privalov's aroach was also the starting oint for E. Quak and the rst named author to investigate in [4], [5] and [6] articular cases of trigonometric wavelet saces. A more general concet of eriodic wavelets was given by G. Plonka and. Tasche in [], where both constructive and eriodized versions are included. The resent aer is organized as follows. In Section we introduce samle saces and discuss dierent bases and their transformations. Beside the interolatory basis, its dual basis and the orthonormalized basis, which all consist of translates of a single function, we also consider a frequency basis. We conclude this section by general L -stability arguments and asymtotic estimates for the basis functions, which are connected to aroximation estimates for the corresonding roections. In Section 3 we restrict ourselves to certain arameters which allow nested samle saces and the construction of wavelet saces. Then the ideas of Section concerning scaling function bases are adated to the corresonding wavelet bases, their transformations and asymtotics. Finally, the reconstruction and decomosition formulas are develoed in Section 4 for the interolatory and orthonormal bases, resectively, in order to rovide the relevant algorithms for signal analysis.

3 Trigonometric Wavelets 3 x Samle saces We denote by C the sace of all continuous -eriodic functions with the maximum norm, by L the sace of square-integrable -eriodic functions and by T n the set of all trigonometric olynomials of degree at most n. The inner roduct in L is given by hf gi Z f(x) g(x) dx for all f g L At rst we consider interolation at equidistant nodal oints in the interval [ ) in its simlest form. This can be achieved by a discretization of the Fourier sum oerator S n with kernel D n, n, S n f(x) a n (a k cos kx b k sin kx) hf D n ( x)i (.) k with the coecients a Z f(t) dt a k f(t) cos kt dt b k Z Z f(t) sin kt dt The Dirichlet kernel D n (x) n s cos sx 8 >< > sin(n ) x sin x n for x Z for x Z k is vanishing at all nodal oints, for k n. Taking corresonding translates of D n as fundamental olynomials, we can dene the Lagrange n interolation oerator, for any f C, at n equidistant oints. The disadvantages of this aroach are, rstly, that eriodic multiresolution needs an even (dyadic) number of nodal oints, and secondly, that the kernel has no bounded L -norm, for n, or, in other words, is not suciently local. Both of these roblems can be resolved by using certain de la Vallee Poussin means of D n instead of the Dirichlet kernel itself.

4 4 J. Prestin & K. Selig. Trigonometric interolation We dene the de la Vallee Poussin means ', for and, by ' (x) m ` D (x) D m (x) cos `x k ` ` cos `x (..) k cos( k)x (..) By induction, one easily roves that they can be rewritten in the form ' (x) 8 >< > sin x sin x sin x for x Z for x Z which guarantees zero values at the nodes k k. (..3) ' ' ' ' 4 Figure. Interolating samling functions The corresonding trigonometric interolation oerator L C C is dened by L f(x) s s f ' x s (..4)

5 Trigonometric Wavelets 5 satisfying, for all k Z, L f k f k For s, the translates ' s(x) ' x s of the de la Vallee Poussin means are linearly indeendent since ' s k ks, for k s, but they are not orthogonal (see Proosition ). Let V san ' s s Then the dimension of V is indeendently of. Being an interolation oerator, L C V is the identity on V. It is instructive to know how single frequencies are maed by L. Lemma. For all k, let k ` mod, with `. Then, L cos(k )(x) 8 cos `x >< > L sin(k )(x) 8 sin `x >< > if ` ( `) cos `x ( `) cos( `)x if < ` < cos( `)x if ` if < ` ( `) sin `x ( `) sin( `)x if < ` < sin( `)x if ` Proof We use the well-known fact that, for all k Z, s s cos x ks sin x ks ( cos x if k mod if k 6 mod ( sin x if k mod if k 6 mod (..5) (..6)

6 6 J. Prestin & K. Selig By denition of L, we have, for any k, L cos(k )(x) cos k( s ) ' (x s ) 4 s kmod 4 cos ks kmod s m m m r s m m r m r cos(rx (k r)s cos r(x s ) ) cos(rx (kr)s ) ( (k r)mod (kr)mod ) cos rx ow we have to consider three dierent cases for `, with k ` mod. If `, then L cos(k )(x) ` If < ` <, then L cos(k )(x) If `, then L cos(k )(x) () ` ` m r m r m m r m r` cos rx cos `x ( r` r `) cos rx cos `x ( `) cos( `)x cos `x ` cos( `)x m r ` cos rx cos( `)x The formula for the sine frequencies is roved analogously. From Lemma, it follows immediately that L is also the identity on T, and T V T (..7) which is basic for nesting subsaces to form a trigonometric multiresolution analysis. A more detailed descrition of V in terms of frequencies is given by the following result.

7 Trigonometric Wavelets 7 Theorem. For <, the set % k k with % (x) % k (x) cos kx % k (x) sin kx (..8) where k, and % (x) cos x % k(x) k % k(x) k k cos( k)x cos( k)x (..9) k sin( k)x sin( k)x where k, constitutes an orthogonal basis of V. Proof By (..7), the orthogonal basis (..8) of T is a basis of V \ T. In order to nd the remaining orthogonal basis of V T, where denotes the orthogonal dierence, we rewrite the high-frequency art of (..) for the translates ' ` as follows k cos ` k k k cos ` cos( k) x ` cos x k k cos (k)` cos( k)x sin (k)` sin( k)x cos ( k)` cos( k)x sin ( k)` sin( k)x cos x k sin ( k)` k cos ( k)` k k cos( k)x k cos( k)x k sin( k)x sin( k)x Thus, we see that the functions in (..9) are elements of V and clearly orthogonal to all functions of T. The mutual orthogonality of these functions is obvious, and their number is dimv dimt.

8 8 J. Prestin & K. Selig Let us introduce vector notations for the sequence of basis functions, ' ' ` ` and % % k k Then the corresonding basis transformations can be described by their transformation matrices. Theorem. The translates of the de la Vallee Poussin means satisfy the relations ' U % and % U ' where the matrices U u Ǹ k and U u k` have the entries, `k k` for all `, u Ǹ u Ǹ k u k` u Ǹ u ` ( )` u ` ( )` 8 >< > cos `k if < k < sin `k if < k < Proof From (..) and the denition of the basis elements % k, it follows immediately that ' `(x) cos `k k % k (x) k sin `k % k (x) which conrms the entries of U. The second relation is determined by the inverse transformation. From the interolation % k(x) L % k(x) ` % k ` and from Lemma, we obtain the entries of U. ' `(x)

9 Trigonometric Wavelets 9. Gram matrix and dual basis For a more detailed investigation of the samle saces, we introduce the Gram matrix G ' r ' s rs To simlify the further comutations, we will emloy the articular structure of this Gram matrix. It is easily seen that G is symmetric and circulant. The latter fact allows us, according to [4], Cha. 3, to diagonalize G by means of the -th Fourier matrix i.e. G F D F where F e irs rs F (F ) (..) Clearly, the diagonal matrix D diag d r contains the eigenvalues of r G. Lemma. The eigenvalues of the Gram matrix G d r 8 < are ( r) if < r < otherwise. Proof We comute the elements of the diagonal matrix D F G in the form d r k h' ' ki e ikr F Since G is symmetric, the eigenvalues are real-valued, and we obtain by the interolation formula (..4) d r k h' ' k h' k cos kr i cos kr ' k i h' L cos(r )i

10 J. Prestin & K. Selig Using Lemma and the reresentation (..) of ' cases. If r, then, we consider again three if r, then and if < r <, then d r d r hd cos(r )i d r hd cos(( r) )i * ` ` ( r) cos(` ) cos(r ) ( r) cos(( r) ) ( r) 4 ( ( r)) 4 ( r) The functions ' k being non-orthogonal, we are now interested in the dual functions. They are needed in order to nd the orthogonal roection of a function f L with resect to the interolating basis. The dual functions ~' r V of the basis functions ' ` (r ` ) are uniquely determined by the orthonormality conditions ~' r ' ` r` As the duals are also in V, they ossess an exansion in V. Hence, let ~' r(x) s rs ' s(x) (..) To simlify the main result on the dual functions, we denote ~' ~' r r Theorem 3. The dual functions satisfy the transformation equations ~' G ' and ' G ~' (..3)

11 Trigonometric Wavelets where G F D F with rs ( )s r G F D F D rs ( )s r diag d r r k k k cos k(s r) rs k cos k(s r) D the entries of which are known from Lemma. Proof From the denition in (..), it follows that ~' r ' ` s rs diag d r r rs ' s ' ` r` (..4) which establishes the coecient matrix. rs G rs Furthermore, the inverse of a circulant is again a circulant, and hence, with D rs G diag d r ` ` d ` sr sr ( )s r F D F r e i`(s r) e i`(s r) k k. Therefrom we calculate ` ( `) e i`(s r) k e i(k)(s r) k cos k(s r)

12 J. Prestin & K. Selig Inverting this relation yields the second formula, and we can comute h' r ' s i d ` e i`(s r) ` ` sr sr ( )s r e i`(s r) k k ` ( `) e i`(s r) k e i(k)(s r) k cos k(s r) For the entries of G, we also know another form from the aer [8] of A. A. Privalov. Proosition. ([8]) Suose <. Then ' r ' r 3 and, for r 6 s, ' r ' s ( ) r s cos (r s) sin (r s) 8 sin 3 (r s) (r s) sin cos (r s) In Section.4 we will use the fact that G is ositive denite. In articular, we roved the following inequality. Proosition. ([7]) For, with, and for r, ' r ' r s s6r ' s ' r > 4 Let us end with the reresentation of the dual functions in the orthogonal basis of frequencies. Theorem 4. For the dual functions, we have ~' U D % with U U from Theorem and D and % D U ~' D as given in Theorem 3.

13 Trigonometric Wavelets 3 Proof We insert the coecients of (..4) in (..) and obtain ~' r(x) ' r(x) k D x r k k k Thus, the dual functions are ~' r (x) k k cos( k) x r D x r k k k By the denition of % k, we can conclude that ~' r k d k s cos (k)(s r) cos( k) x r cos( k) x r k k cos( k) x r cos rk % k d k k sin rk ' s (x) % k (..5) Writing this in matrix notation yields the rst equation. Then the inverse transformation works by means of the identity (U (D ) ) D U From (..5), it follows that the dual functions are shifts of ~' again, which is an even function like '. ote also that the coecients rs are equal to the function values rs () ~' (s r) (..6) which follows from the interolation roerty of the basis in the exansion (..)..3 Orthonormal basis of translates We also seek a basis of V consisting of orthonormal translates O' r (x). Writing O' x r of a function O' V O' r(x) s rs ' s(x) (.3.)

14 4 J. Prestin & K. Selig for all r, we have to determine the coecients rs. The coecient matrix rs rs () O' (s r) rs is obviously circulant. Hence, we can write F F, with a diagonal matrix. If we require the orthonormality O' k O' r and denote ` s O' k` rs ' ` ' s kr (.3.) O' r r then we can summarize the reresentation as follows. Theorem 5. In V, there exists a basis of orthonormal translates of an even function O' satisfying the relations where O' ' ' O' (.3.3) O' ~' and ~' O' (.3.4) F F rs ( )s r k k cos k(s r) rs (.3.5) and F F with rs ( )s r k q diag d ` ` and d ` from Lemma. k cos k(s r) rs q diag d ` `

15 Trigonometric Wavelets 5 Proof With q diag d ` `, we are able to comute the coecients rs, for r s, from the equation and obtain rs e i`(s r) ` ` q d ` sr sr ( )s r The entries of sr e i`(s r) k k sr ` ` sr ( )s r sr qd ` e i`(s r) e i`(s r) k ` d ` F F e i`(s r) k e i(k)(s r) k cos k(s r) F k F follow from k e i(k)(s r) k cos k(s r) ow we rove the orthonormality of the translates for the reresentations in (.3.3). From (.3.5), we derive that rs sr. Hence, O' is even, and we have G T G F D F I being the -th identity matrix, which roves (.3.). The relations in (.3.4) are easy to derive from (.3.3) and (..3). Finally, we want to reresent the orthonormal translates in the basis of frequencies. Theorem 6. For the orthonormal translates, we have the relations O' U % and % U O' (.3.6)

16 6 J. Prestin & K. Selig with U U from Theorem and as given in Theorem 5. Proof We obtain an exlicit form for the orthogonal translates of O' by evaluating with Thus, O' r(x) s ' r s ' r rs' s(x) k cos (k)(s r) s k k k cos (k)(s r) k s cos (k)(s r) ' s (x) L cos(( k) ) x r cos( k) x r O' r(x) D (x r ) This can be rewritten O' r k k k q cos kr d % k k which yields the matrices in (.3.6). k cos( k) x r ' s (x) ' s (x) k cos( k)(x r k ) q d k sin kr % k.4 Time-frequency-localization and roection estimates L -orms and stability Our rst aim is to describe the time-localization in terms of the L -norms of the scaling functions. Therefore let us dene, for <, kfk Z f(x) dx and kfk ess su f(x) x[]

17 Trigonometric Wavelets 7 and analogously for vectors f k g C kf k gk` k k and kf k gk` max k k From the well-known inequalities of Holder and ikolskii (see e.g. A.F. Timan [], Cha. 4.9), one knows that kt nk q kt n k C n q for all olynomials t n T n and q. We want to have basis functions with the order n q on the right-hand side, which ensures the best ossible time-localization with resect to a olynomial RA. In articular, we are looking for olynomials satisfying kt n k n kt n k nkt n k The notation A n B n means A n B n C and A n Bn C. Here and in the following, C denotes ositive constants deending on xed arameters involved, but their values may be dierent at each occurrence. Let us recall the basic result of an L -estimate for the interolating functions established by S.B. Steckin [] k' k log C O In order to obtain exlicit constants, we roceed with the following simle derivation. For, k' k < < Z sin(t) sin(t) sin t dt Z 4 sin(t) Z sin t dt sin t dt 4 ln 3 ln (.4.) This comutation contains some information about the time-localization of ' in the three dierent intervals of integration. oreover, we know the maximum value of ', and thus k' k (.4.) In Section 3. we will dene and in such a way that 3 and that is bounded from above by a constant. Assuming to be constant,

18 8 J. Prestin & K. Selig we conclude from (.4.) and (.4.) by means of ikolskii's inequality, for arbitrary, that k' k (.4.3) As a further ste, we formulate the Riesz stability in the more general L - setting. Theorem 7. For arbitrary sequences f k g C and, the inequalities kf kgk` () k k ' k A kf k gk` (.4.4) are satised, with indeendent of, or A A < k' k C if < < C if < (.4.5) As noted above, for simlicity, we assume 3 K (.4.6) where K is a constant indeendent of. This imlies the uniform boundedness of A in (.4.4), for, because of A 4 3 (3 ln ) < 5 ln K 3 Proof a) We start with the inequality on the left-hand side, which can be roved by using a well-known inequality for trigonometric olynomials t n T n (see e.g. [], Cha. 4.9), namely su x ` t n x ` n kt n k (.4.7)

19 Trigonometric Wavelets 9 for and. Then, by alying the interolatory roerties of ' k, we conclude that ` ` 4 3 ` ( ) k k ' ` k k ' k k k ' k k b) In order to rove the inequality on the right-hand side, we use, for <, the function g from the sace L q, where q, with kgk q such that k k ' k Then Holder's inequality gives k k Z Z g(x) ' k(x) dx k g(x) k k k where denotes the usual -eriodic convolution, g ' (t) Z k ' k (x) dx (g ' ) k q g(x) ' (t x) dx ow we aly (.4.7) and obtain two dierent estimates, k (g ' ) k q 8 >< 4 3 > q 4 3 kg ' k q q kgk q k' k if q < m ks m gk q if < q <

20 J. Prestin & K. Selig The second estimate in (.4.5) follows from the boundedness of the Fourier sum oerator S m in L q, for < q < (see e.g. [5]). The roof is comleted by a similar derivation for. Alying (.4.7) again, we nd that k ' k max k k k su x kf k gk` 43 3 k k' k Asymtotics for the dual and orthogonal functions ' x k Our next goal is to achieve the same time localization results for the dual functions ~' and for the orthogonal functions O' as we have in (.4.3) for the interolatory functions. At rst we determine estimates for the coecients rs and rs. Lemma 3. The coecients in the basis reresentations (..) and (.3.) satisfy the decay conditions, for r s, rs rs C max (r s ) ( r s) (.4.8) Proof For r s, both of the coecients given in (..4) and (.3.5) are less than, since <. For r 6 s, let us focus on rs k k We dene the -eriodic even function f by f (x) 8 < cos k(r s) () x for x for < x The derivative of f is of bounded variation and ossesses two ums in [ ]. Hence, for the Fourier coecients a r s, with r s, we have (see [5], Cha. ) ar s Z f (x) cos(r s)x dx C(r s ) and by the aliasing formula for discrete Fourier coecients, rs 4 k ` f ( k ) cos(r s) k a `r s a ` rs C max (r s) r s

21 Trigonometric Wavelets Thus, (.4.8) is roved for rs. The same idea works for rs (see Theorem 3), if one relaces f by f (x) 8 < x for x for < x ow it is easy to obtain the same time-localization result for the dual and orthonormal functions as we have in (.4.3) for '. Theorem 8. For, we have the asymtotic behaviour k ~' k ko' k Proof The roof simly consists of the alication of Lemma 3 to (.4.4). E.g., we obtain ko' k k' k k k k Let us end this subsection with a quite dierent descrition of time localization (see also [6]). In [ ], we dene, for k, the intervals (k ) I k k and describe the decay in terms of the distance from the eak. Again, we start with the interolating function ' (x). From (..3), one obtains by standard estimates and (.4.6) that 8 >< max ' k if < k (x) (.4.9) xi k > if < k < ( k) ow, for simlicity, we restrict ourselves to < k. The same time localization results can be shown for the dual and orthonormal functions. Theorem 9. Let < k. Then k ~' k ko' k (.4.) and max xi k ~' (x) O' (x) C k (.4.)

22 J. Prestin & K. Selig Proof Formula (.4.) is already included in Theorem 8. Here we only show the estimate in (.4.) for O'. With Lemma 3, (.3.) and (.4.9), we obtain max xi k O' (x) C max xi k r C r r (r ) ' r (x) ((r )(r k )) r ( r) ' r (x) ( r) (minf(r k) ( r k)g) Then (.4.) follows directly by slitting the sums on the right-hand side into Error estimates r k r 3k rk r3k Here we want to summarize aroximation roerties of the interolatory and orthogonal roections onto the saces V. For simlicity, we only deal with the most interesting case of continuous functions and estimates in the sunorm. Roughly seaking, the same results are available for L and esecially for L < <, avoiding the logarithmic term in the estimates. In articular, we want to comare the aroximation order with the best aroximation by trigonometric olynomials of corresonding degree, as A. A. Privalov mentioned in [8]. Further results connecting the convergence order of a given function f with smoothness conditions on f can be easily deduced. Denote, for f C, E n (f) inf fkf t n k t n T n g In this subsection we do not necessarily assume the uniform boundedness of as done in (.4.6). But, for further use, we restrict ourselves to However, in the case of (.4.6), we obtain error estimates in the order of the best aroximation. Theorem. Let f C. For the interolatory roection L onto V have the estimates, we E (f) kf L fk (A ) E (f) (.4.) with A as dened in (.4.5).

23 Trigonometric Wavelets 3 Proof The inequality on the left-hand side follows immediately from L f T For the inequality on the right-hand side of (.4.), we write E (f) kf t k to conclude with the hel of (.4.4) that kf L fk E (f) kt L fk E (f) A kt fk ow we are looking for error estimates for the orthogonal roection. The roof of Theorem shows that we only have to deal with the oerator norm of the orthogonal roection oerator P L V, P f k hf ~' ki ' k k hf O' ki O' k Lemma 4. The orthogonal roection P onto V satises Proof Writing we obtain by (.4.4) kp k CC P f k C log k ' k (.4.3) kp fk C k' k max k< k C k' k ` Taking the inner roduct with ' ` in (.4.3), we estimate hp f ' `i Alying Proosition yields Hence, k k h' k ' `i ` h' ` ' `i k k6` h' k ' `i ` 4 hp f ' `i 4 hf ' `i 4 kfk k' `k kp fk C k' k kfk which, together with (.4.), roves the lemma. Using Lemma 4 and standard roection estimates, we arrive at the following nal result.

24 4 J. Prestin & K. Selig Theorem. Let f L. For the orthogonal roection P onto V, we have the estimates E (f) kf P fk C log E (f) (.4.4) It remains an oen question whether one can relace the factor log in (.4.4) by log, the best ossible constant in (.4.). x3 3. Trigonometric RA Wavelet saces For eriodic olynomial multiresolution analyses, it is natural to consider a dyadic number of nodal oints (see []). To allow more generality for ractical alications, we additionally allow any (odd) factor c and dene ( if c if < (3..) for all and any constant. trigonometric RA fv g of L by For xed c and, we dene a V V Theorem. If c 3, then all roerties of a eriodic RA are satised, namely (i) V V for all dimv [ (ii) clos L V L (iii) f V ) f k V for all k Z (iv) 9 V V sanf k k k g Taking ' as the generating function, for any and f k g C, we have the stability condition A kf k gk` k k k with best ossible constants A and B. B kf k gk` (3..)

25 Trigonometric Wavelets 5 ote that, if we relace k by the orthonormal translates O k O' k, then we obtain best stability i.e. A B. Proof The inclusion relation V V, for all, follows from (..7), if For <, this is obviously true, and for, it means that This yields the condition 3 c 3 c i.e. 3 for all Then we have, which, by (..7), imlies roerty (ii). The third and fourth roerties follow directly from the construction in Section.. It remains to establish the stability constants in (3..). Let us denote by v r, for r, the eigenvectors of the Gram matrix G (h k `i) k` such that G v r d r v r, where d r d r are the eigenvalues known from Lemma. The eigenvectors form an orthogonal basis of C. Therefore we can write any -dimensional vector as a linear combination P r a r v r, with comlex coecients a r. Let P h i E k k k denote the Euclidian inner roduct of C. Then we have * k k and since k k k hg i E k h i E P k k k k k. P l r r l l d r a r hv r v r i E a r hv r v r i E the quotient k k is bounded by the minimal and maximal eigenvalue of G, resectively. Let L L denote the interolation oerator maing C onto V. We can rove the following two-scale relations for the shifted scaling functions.

26 6 J. Prestin & K. Selig Theorem 3. For r, we have the renement equations r (x) r (x) and, for, the dilation relation s (x) (x) cos x (s ) r s (x) (3..3) (3..4) ote that in (3..4) the function ( cos x) maintains the values of ( ) around k and suresses the interosed oscillation of ( ) around (k), where k Z. Proof By the interolation roerty of r, we have r (x) L r 4 k r r (x) s k (x) A (s) r s (x) A For all levels, we have and, which yields (x) sin (x) sin (x) 4 sin x (x) cos x cos x Furthermore, we need the following two-scale inner roduct. Lemma 5. For `, and m, we have h ` m i ` m (3..5) Proof By (.), (..) and hcos(m ) cos(n )i mn, we can write h ` m i D ` D ` m E m

27 Trigonometric Wavelets 7 3. Interolating wavelets The wavelet saces are dened as the relevant orthogonal comlements of V in V i.e. V V W, for all, where denotes the orthogonal sum. From Theorem, we derive a simle orthogonal basis of the wavelet saces. Theorem 4. An orthogonal basis of W is given by the set with the functions f ` ` g (x) cos x (x) sin x k(x) cos kx k (x) sin kx (3..) where k k (x) k k(x) k cos( k)x k sin( k)x k cos( k)x (3..) sin( k)x where k and k(x) k k (x) k cos( k)x k sin( k)x k cos( k)x sin( k)x (3..3) where k. Proof First, we check that all functions `, for `, belong to V. By Theorem, this is obvious for the olynomials in (3..) and (3..3). Let ` % `. In (3..), we have, for k, k k k k Second, we verify that h ` k i, for all k `. This is evident for all ` in (3..) and (3..). For (3..3), it follows by simle calculations. Regarding shift-invariant saces, we are now looking for a localized wavelet function V that generates the wavelet sace W i.e. W san n n

28 8 J. Prestin & K. Selig To determine the wavelet uniquely, we may additionally require interolation roerties for the translates n, in this case at the interolation nodes of V which are not interolation nodes of the translates of. Hence, we demand that V, h k i for all k (3..4) and, in analogy to the scaling function, that (m ) m for all m (3..5) Theorem 5. There exists a uniquely determined function V satisfying (3..4) and (3..5), namely (x) x x (3..6) For its translates n (x) (x n ), n, the renement equations are given by n(x) n (x) s For, we have the secial dilation equation (x) (x) cos(x Proof For an arbitrary f V, there is an exansion f L f (n ) s s (x) (3..7) 4 s ) s s (3..8) with 4 coecients s f( s ). By means of (3..5), the orthogonality condition in (3..4) is equivalent to h k fi 4 s h k s i s 4 s s k s

29 Trigonometric Wavelets 9 for all k. The interolation roerty in (3..5) holds if and only if (m ) f m m for all m. Hence, f k k k is the only function in V satisfying (3..4) and (3..5). By (3..3), it follows f, where is dened in (3..6). For its translates, (3..7) is obtained, using that is an even function. Finally, by means of (3..4), we can write, for, (x) x x which roves (3..8). x x cos(x ) Figure. Interolating wavelet functions

30 3 J. Prestin & K. Selig ow we can dene the interolation oerator R C W, R f (k ) f k k (3..9) which is the identity in W and fulls, for k Z, R f (k) f (k) Further, we are interested in the exansion of the localized basis in the orthogonal basis from Theorem 4. Let us introduce the notations for the vectors of the functions ( r ) r and ( k ) k Theorem 6. The translates of the wavelet dened in (3..6) satisfy the relations O and O where the matrices O (o rs ) rs, the entries and O (o sr ) sr have, for r o r o r o r ( )r o r ( )r o rs o sr 8 >< > (r )s cos if < s < (r )s sin if < s <

31 Trigonometric Wavelets 3 Proof Similar to (..), we can write r(x) D x (r) k k k cos( k) k cos( k) k ` k cos ` D x (r) x (r) x (r) k cos( k) x (r) x (r) k cos( k) x (r) Rewriting this in the basis dened in Theorem 4 gives r(x) k k cos ( k)(r ) sin ( k)(r ) k (x) k(x) which yields the entries of O. The inverse transformation matrix O is easily obtained by the interolation s (x) R s (x) r (r ) s r(x)

32 3 J. Prestin & K. Selig 3 ( 8 ) 3 ( 8 ) 3 Figure 3. Frequency sectrum of and (k ) k( ) k k 3.3 Gram matrix and dual wavelets In the sequel, the index of a matrix indicates that its dimension is. In articular, let I be the -th identity matrix and F be the -th Fourier matrix dened in (..). We denote the Gram matrix of the translates of the wavelet function W by H (h r s i) rs The Gram matrix is symmetric and circulant, and so we can diagonalize it as well by means of the -th Fourier matrix F such that with a diagonal matrix E diag (e k ) k. H F E F (3.3.) Lemma 6. The eigenvalues of the Gram matrix H are e k 8 >< > k if k < if k ( k) if < k < if k ( k) if < k

33 Trigonometric Wavelets 33 Proof By means of (3..6) and (3.3.), it is ossible to comute the eigenvalues as elements of the diagonal matrix E F H F, using that Hence, h r s i where G D r r s s E h r s i h r s i D h r s i h r s i rs (h r s i) H G G I rs E (s r) G and G (h r s i) are known to be circulant. By the distributive law for matrices, we have H F E F F D D I F with D diag d r r rs and D diag ( `) `. To comute the elements of D, we rewrite the elements of the circulant G, ` Then, by Lemma, we obtain ` e i`(s r) h r s i ` d ` d ` 8 >< > 3 ` 4 4 ` ` ` d ` e i`(s r) d ` e i`(s r) d ` d ` if ` < if ` 3 ( `) 4 e i`(s r) if < `

34 34 J. Prestin & K. Selig The eigenvalues of H follow immediately, for k, from e k k d k ow, we dene the dual functions ~ r W of the interolating translates `, for r `, by the orthogonality conditions D ~r `E r` (3.3.) As elements of W, they have a reresentation ~ r (x) We denote the function vector ~ s rs s (x) (3.3.3) ~r r Theorem 7. For the dual functions, we have the relations with and H ~ H H F E F F rs F rs and k k k k cos k H ~ ( k)(s r) k k cos (k)(s r) cos k(s r) A rs k cos k(s r) k A rs where E diag (e r ) E r diag (e r ) r the entries of which are known from Lemma 6. (3.3.4)

35 Trigonometric Wavelets 35 Proof By (3.3.) and (3.3.3), we have D ~r `E rs h s `i r` which yields s ( rs ) H rs F E F, we can com- Due to the circulant structure of the inverse H ute its elements as rs k e ik(s r) e k k sr e ik(s F E F k k k e ik(s r) A k k cos (k)(s r) k k cos k(s r) k k A Finally, we need the entries of H F E F, which are h r s i k e k e ik(s r) k sr e ik(s r) k k k k k i( k)(s r) e A k cos cos k(s r) ( k)(s r) A i( k)(s r) e e ik(s r)

36 36 J. Prestin & K. Selig A. A. Privalov comuted the entries of H as follows. Proosition 3. (see [8]) If i.e., and if 3, then and, for r 6 s, h r s i 3 (r s) 4 sin cos (r s) h r r i sin3 (r s) cos (r s) sin (r s) 8 ( ) r s cos (r s) 4( ) r s sin (r s) For comleteness, we deduce the reresentation of the dual translates in terms of the frequencies. Theorem 8. We can transform ~ O E and E O ~ with O O from Theorem 6 and E E as given in Theorem 7. Proof First, we comute the dual wavelets from (3.3.3), ~ r (x) r (x) k k k k cos s s ( k)(s r) cos k(s r) s(x) s(x)

37 Trigonometric Wavelets 37 k ` k k cos( k) x (r) cos ` x (r) k cos( k) x (r) k k k k k cos( k) k cos( k) k x (r) cos( k) x (r) k cos( k) x (r) x (r) and therefore ~ r (x) ` k cos ` k k cos( k) x (r) k x (r) k cos( k k) x (r) Therefrom the transformation matrices follow easily. ote that ~ r(x) ~ (x r ), for r, are translates of one function, again. oreover, since V? W, we have D ~r s E for all r s and. D ~r s E

38 38 J. Prestin & K. Selig 3.4 Orthonormal translates In order to determine wavelets with orthonormal translates, we roceed in the same way as for the scaling functions i.e. via the Gram matrix. We need orthonormal translates O r (x) O x r of a function O W i.e. coecients rs, such that and O r (x) s rs s (x) ho r O k i rk (3.4.) for all r k. As in Section., we use the matrix notation and introduce the coecient matrix V ( rs ) and the function vector rs O (O r ) r with the rela- Theorem 9. There exists an orthonormal wavelet basis of W tions O O V V V ~ and ~ V O O (3.4.) (3.4.3) where V F rs V F F rs ( )s r k k k k cos k(s r) cos (k)(s r) A k cos k(s r) k k k cos k(s r) rs rs (3.4.4) (3.4.5)

39 Trigonometric Wavelets 39 with diag e ` and ` the entries of which are given in Lemma 6. diag e` ` Proof Here we follow the same line as in the roof of Theorem 5. At rst, we show (3.4.4) and (3.4.5). With V F F ( rs ), we comute rs k e ik(s r) ek e ik(s r) k Then, for the entries of we have sr V sr k k k e k rs e ik(s r) k k k cos F F ( sr ) sr ek e ik(s r) e ik(s r) k k k k ( )s r k k k k i( k)(s r) A k cos k(s r) ( k)(s r) A i( k)(s r) e A cos k(s r) e ik(s r) cos k(s r)

40 4 J. Prestin & K. Selig So, V and V are symmetric. Hence, t he orthonormality of the translates O r, required in (3.4.), follows easily from V H V T F E F I This, together with (3.4.) and Theorem 7, roves (3.4.3), too. Last but not least, we give the reresentation of the orthonormal translates in the basis of frequencies. Theorem. We have O O and O O (3.4.6) with O O from Theorem 6 and as given in Theorem 9. Proof First, we need an exlicit formula for O (x) s (x) s s (x) k k k cos k s s ( k)s cos ks s(x) s(x) Therefore we consider, for < k < and < k <, the sum s cos ks s(x) s cos ks s cos ks s x s x

41 Trigonometric Wavelets 4 By means of (..5) and (..6), we calculate the rst art, s cos ks s (x) cos ks s k cos `x s 8 < (k ` ` cos ` x s ` cos( `) s ` cos ` cos ( `)x (k `)s k ` ` ` `x (k `)s x s A ( `)x (k`)s (k `)mod (k`)mod cos `x A (k `)mod (k`)mod cos( `)xa cos kx k cos( k)x if k < (cos kx cos( k)x) if k < The second art is known from Lemma. Hence, cos ks 8 >< > s s(x) k cos( k) ( k) x k cos( k) x cos k x ( k) cos( k) if < k < x if < k <

42 4 J. Prestin & K. Selig Consequently, O (x) ` k k k k cos ` x k cos( k) x k cos( k) x k k k cos( k) cos( k) x ( k) k cos k ( k) cos( k) x x x and nally, O (x) k ` k k cos( k) x k cos ` x k cos( k k) x From this, the matrices in (3.4.6) can be easily derived. 3.5 Localization and roection estimates We resent results for the wavelet saces, which are analogous to the ones of Section.4. Here, we can be brief because most of the roofs of that section can be adated easily.

43 Trigonometric Wavelets 43 L -orms and stability As in (.4.6) in Section.4, we assume to be bounded from above by a constant. Then the L -stability of the wavelets W, for, is given by k k ( ) kf k gk` k This can be roved by alying (3..6) and (.4.4). For the secial case k k, we obtain immediately the same asymtotics as we have in (.4.4), namely k k ( ) for any In order to determine the asymtotics of the dual and orthonormal wavelets, we follow the same line as in Section.4. Thus, we have to estimate the coecients rs and rs. Lemma 7. For the entries of the circulant matrices H and V, the inequalities rs rs C max (r s ) ( r s) hold, for r s. Proof Again, we have to estimate discrete Fourier coecients. But this time, with resect to (3.3.4) and (3.4.4), we consider the -eriodic functions f (x) and f (x) 8 >< > 8 >< > x x ( ) s r x x x x for x for < x for < x ( ) s r x ( )s r for x x x for < x for < x which have a iecewise continuous rst derivative. The alication of Lemma 7 gives the time-localization result for the dual and orthonormal wavelets.

44 44 J. Prestin & K. Selig Theorem. For, k ~ rk ko r k ( ) The ointwise estimates, which corresond to (.4.9) and (.4.), can be deduced using (3..6). Theorem. For simlicity, we restrict ourselves to < k. Then n o (x) O (x) ~ (x) max xi k Error estimates C k In Section.4 we investigated the aroximation of continuous functions by elements from the samle sace V. ow we consider the corresonding results for aroximation rocesses in W. Of articular interest are the interolation oerator R dened in (3..9) and the orthogonal roection Q that can be easily handled by Q P P Here, we only assume (3..) for and, with arbitrary. Theorem 3. For the interolatory and orthogonal roections, we have kr k CC C log and kq k CC C log ote that this Theorem is a key to nd olynomial bases of C. We discuss this in the aer [7]. x4 Decomosition and reconstruction The wavelet analysis of functions is based on the transformations between a suciently large level samle sace and the wavelet saces of lower levels i.e. the iterative decomosition of V into the orthogonal sum V W. Starting from an aroximation of a given function in an secic samle sace, either by interolation or by orthogonal roection, we only have to know the corresonding basis transformations to calculate the wavelet coecients of the function. Conversely, we can reconstruct the roection of the function from the wavelet coecients. 4. Interolating bases For the interolatory scaling and wavelet functions, we have already determined the renement relations in the Theorems 3 and 5. ow we resent the reverse basis transformation.

45 Trigonometric Wavelets 45 Theorem 4. The decomosition formulas, for `, are `(x) s ~ s ` s (x) s (s ) ~ ` s(x) (4..) and ` (x) s s (` ) ~ s s (x) ~ ` s s(x) (4..) Proof We only have to calculate the coecients a `s and b `s, such that `(x) s a `s s (x) s b `s s (x) for ` 4. These coecients are the inner roducts a `s D ` ~ s E D ` ~ s ~ s `

46 46 J. Prestin & K. Selig and b `s D E ` ~ s D D ` k k k cos( k) ` k k cos( k) k cos k k cos k(` s) k ( )` ( )` ( k) k k k ~ s (` ) 8 >< > ~ ` ~ ` (s) (s) k cos( k k) (s) k ( k)(` s) cos k cos ( k)(` s) A cos k(` s) cos k(` s) k k k cos ( )` s (s) ( k)(` s) A ~ (s ` ) for odd ` for even `

47 Trigonometric Wavelets 47 otivated by the dierent structures for translates ` with even and odd index `, resectively, we introduce a ermutation matrix P that yields C A rs rs rs P (4..3) T In order to write the basis transformations in matrix-vector-notations, analogous to Sections and 3, we have to investigate the following matrix of function values. Lemma 8. The singular matrix K r (s ) rs F Q F with the diagonal matrix Q diag (q r ) r, has the eigenvalues q r e ir 8 >< > and the transose K T F Q F. Proof From K if r r if < r < if r ((s r)) matrix is circulant and can be written as K rs, we conclude that the F Q F, where Q is a

48 48 J. Prestin & K. Selig diagonal matrix. The entries of Q follow from q r (s) s s L cos(r ) cos r s e irs s i i L sin(r ) s and the alication of Lemma. Since K is real-valued, K T K T F Q F sin r s s ow we can formulate the decomosition and reconstruction in matrix notation to be used in numerical algorithms. Theorem 5. Let A and B be the transformation matrices in the decomosition and the reconstruction equations such that P T B and Then A and B have circulant blocks, namely A I F Q F F Q F I F D F F D Q F F D Q F F D F A P A with Q known from Lemma 8 and D D as given in Lemma. Proof From the relations (3..3) and (3..7), together with (4..3), we derive the matrix A I K K T I r s s (r) rs rs (s) r rs r s rs C A

49 Trigonometric Wavelets 49 where K is the matrix described in Lemma 8. Analogously, we deduce from (4..) and (4..) that G ~ r s ~ s (r) G G K T G using (..6) and Theorem 3. rs rs K A r ~ (s) rs ~ r s rs C A 4. Orthonormal bases For orthonormal bases, the transformation matrices are unitary. The decomosition and reconstruction relations for the orthonormal translates are given, for ` 4, by O ` ho ` O r i O r r r from which we deduce the transformation matrices. ho ` O r i O r (4..) Theorem 6. For the matrices OA and OB in the relations and we have O P T OB O O O O (4..) OA P O (4..3) OA OB T F F F Q F F Q F F F with as given in Theorem 5 and Q from Lemma 8.

50 5 J. Prestin & K. Selig Proof Analogous to the roof of Theorem 5, we distinguish between the translates O ` with even and odd index `. Then formula (4..) is relation (4..) in matrix notation, with OB ho s O r i sr ho s O r i sr ho s O r i sr ho s O r i sr C A Similarly, we obtain (4..3) with OA OB T. We comute, for ` 4 and r, ho ` O r i D D ` D r D (` r) O r ` k k cos( k) k k r E k (k)(` r) cos k and * ho ` O r i D ` k k k k k cos( k) k k cos( k) k cos k (r) (r) ` k cos( k) (r) k

51 Trigonometric Wavelets 5 k k k 8 >< > k ( k)(` r ) cos k cos k(` r ) ( k) k (r) O r ` O ` cos ( k)(` r ) for odd `, for even `. Hence, from Theorem 5 and Theorem 5 we conclude that ( K ) T K A which nally roves the desired form of the transformation matrices. x5 Conclusion In this aer we have studied nested subsaces V V of trigonometric olynomials constructed from translates of de la Vallee Poussin kernels ' based on averaging over Dirichlet kernels. oreover, we considered orthogonal olynomial wavelet saces W V V and their biorthogonal wavelet bases. The greater the number for xed, the better time-localized those basis functions behave, as it was investigated in Sections.4 and 3.5. On the other hand, the best frequency-localized translates as well as the best frequency slittings between the wavelet saces W are obtained for i.e., which is the original Fourier case at an even number of nodes. Our considerations have been focused not only on the interolatory bases and their dual bases, but also on two orthogonal bases in each sace. Among all orthogonal bases functions in V and W, resectively, O r and O s are the most time-localized (and translation invariant) bases functions, and k and ` are the most frequency-localized ones. The main art of the aer has been concerned with the algorithms for the basis transformations being described by their transformation matrices with resect to fast numerical imlementations. For the circulant matrices in i) and ii) below, the knowledge of the eigenvalues reduces the comutations to the alication of the FFT. In ii), we also need to ermute the vector entries.

52 5 J. Prestin & K. Selig Let us summarize the relationshi between the bases by the relevant matrices in the following schemes i) for the samle saces V or V and the wavelet saces W on one level? O G - ~ G @ - U? %? O H - ~ H @ O V OE @R - O ii) for the decomosition and reconstruction V V W P A - B? OA O P O - O OB Error estimates for the initial aroximation of a given function by the interolatory or the orthogonal roection onto a samle sace of a suciently large level are included in Section.4. Let us end with the remark that r (t) r (t) and r (t) r (t) are even functions. Therefore, this aroach can be transformed by x cos t, for t [ ], to the algebraic case which yields olynomial wavelet saces on [ ] orthogonal with resect to the Chebyshev weight ( x ). For this see [5], [] and []. References [] Chui, C. K., An introduction to wavelets, Academic Press, ew York, 99. [] Chui, C. K. and H.. haskar, On trigonometric wavelets, Constr. Arox. 9 (993), 67{9. [3] Daubechies, I., Ten lectures on wavelets, CBS-SF Series in Al. ath., SIA ublications, Philadelhia, 99. [4] Davis, P. J., Circulant matrices, Wiley Interscience, ew York, 979.

53 Trigonometric Wavelets 53 [5] Kilgore, T. and J. Prestin, Polynomial wavelets on the interval, Constr. Arox., 995 to aear. [6] Lemarie P. G. and Y. eyer, Ondelettes et bases hilbertiennes, Rev. at. Iberoamericana (986), {8. [7] Lorentz, R. A. and A. A. Sahakian, Orthogonal trigonometric Schauder bases of otimal degree for C( ), The Journal of Fourier Analysis and Alications (994) no., 3{. [8] eyer, Y., Ondelettes, Herman, Paris, 99. [9] On, D. and K. Oskolkov, A note on orthonormal olynomial bases and wavelets, Constr. Arox. 9 (993), 39{35. [] Perrier, V. and C. Basdevant, Periodic wavelet analysis, a tool for inhomogeneous eld investigation, theory and algorithms, Rech. Aerosat. 3 (989), 53{67. [] Plonka, G., K. Selig and. Tasche, On the construction of wavelets on the interval, Adv. Com. ath., 995 to aear. [] Plonka, G. and. Tasche, A unied aroach to eriodic wavelets, in Wavelets Theory, Algorithms, and Alications, C. K. Chui, L. ontefusco, and L. Puccio (eds.), Academic Press, San Diego, 994, 37{5. [3] Prestin, J., On the aroximation by de la Vallee Poussin sums and interolatory olynomials in Lischitz norms, Analysis athematica 3 (987), 5{59. [4] Prestin, J. and E. Quak, Trigonometric interolation and wavelet decomosition, umer. Algorithms 9 (995), 93{37. [5] Prestin, J. and E. Quak, A duality rincile for trigonometric wavelets, in Wavelets, Images, and Surface Fitting, P. J. Laurent, A. Le ehaute and L. L. Schumaker (eds.), A K Peters, Wellesley, 994, 47{48. [6] Prestin, J. and E. Quak, Decay Proerties of trigonometric Wavelets, in Proceedings of the Cornelius Lanczos International Centenary Conference, J. D. Brown,. T. Chu, D. C. Ellison and R. J. Plemmons (eds.), SIA, 994, 43{45. [7] Prestin, J. and K. Selig, On a constructive reresentation of an orthogonal trigonometric Schauder basis for C, in rearation. [8] Privalov, A. A., On an orthogonal trigonometric basis, ath. USSR Sbornik 7 no. (99), 363{37.

54 54 J. Prestin & K. Selig [9] Selig, K., Trigonometric wavelets for time{frequency{analysis, in Aroximation Theory, Wavelets and Alications, S. P. Singh (ed.), Kluwer Academic Publ., Dordrecht, 995, 453{464. [] Steckin, S. B., On de la Vallee Poussin sums, Doklady Akad. auk SSSR 8 no. 4 (95), 545{548 (russ.). [] Tasche,., Polynomial wavelets on [ ], in Aroximation Theory, Wavelets and Alications, S. P. Singh (ed.), Kluwer Academic Publ., Dordrecht, 995, 497{5. [] Timan, A. F., Theory of aroximation of functions of a real variable, English translation Pergamon Press, 963. [3] Wotaszczyk, P. and K. Wozniakowski, Orthonormal olynomial bases in function saces, Isr. J. ath. 75 no. /3 (99), 67{9. [4] Wozniakowski, K., Orthonormal olynomial basis in C() with otimal growth of degrees, rerint. [5] Zygmund, A., Trigonometric Series, Cambridge University Press, Second Edition, 959. Jurgen Prestin FB athematik Universitat Rostock D{85 Rostock Germany restin@mathematik.uni-rostock.d4.de Kathi Selig FB athematik Universitat Rostock D{85 Rostock Germany selig@mathematik.uni-rostock.d4.de