On the Stability of Multivariate Trigonometric Systems*

Transcription

1 Joural of Mathematical Aalysis a Applicatios 35, Article ID jmaa , available olie at o O the Stability of Multivariate Trigoometric Systems* Wechag Su a Xigwei Zhou Nakai Istitute of Mathematics, Nakai Uiersity, Tiaji 30007, People s Republic of Chia xwzhou@su.akai.eu.c Submitte by Joseph D. War Receive March 0, 998 Kaec s -theorem says that if : Z is a seuece of real umbers for i which L, the e : Z forms a Riesz basis for L,.S. Favier, R. Zalik, C. Chui, a X. Shi extee this result to the multivariate case. But their results lea to very small stability bous. I this paper, we give a optimal stability bou for the multivariate trigoometric systems. Moreover, for the case of Fourier frames i L,, we also give the stability bous. 999 Acaemic Press Key Wors: Kaec s -theorem; stability bou; frame; Riesz basis.. INTRODUCTION AND MAIN RESULTS A family of fuctios f : j J j belogig to a separable Hilbert space H is sai to be a frame if there exist positive costats A a B such that A f ² f, f : B f jj j for every f H. The umbers A a B are calle the lower a upper frame bous, respectively. If oly the right-ha ieuality is satisfie for every f H, the f : j J is sai to be a Bessel seuece with bou B. j * This work was supporte by the Natioal Natural Sciece Fouatio of Chia ŽGrat No a RFDP. Correspoig author X99 $30.00 Copyright 999 by Acaemic Press All rights of reprouctio i ay form reserve.

2 60 SUN AND ZHOU Family f : j J j is calle a Riesz basis if it is complete i H a there exist positive costats A a B such that j j j j j jj jj jj A c cf B c, c l Ž J.. A frame that ceases to be a frame whe ay oe of its elemets is remove is sai to be a exact frame. It is well kow that exact frames a Riesz basis are ietical Žsee. 7. Kaec s -theorem 6 states that if is a seuece of real umbers for which L, Z, i the e forms a Riesz basis for L,. For the multivariate case, we ca ask a similar uestio. Let,..., Z,,..., R. We wat to fi a costat i, : such that e : Z is a Riesz basis for L, wheever sup sup sup. k k Z Z k We call a stability bou. Favier a Zalik prove the followig propositio 5, Corollary. PROPOSITION.. Assume that k k L, k,...,, a that i, : L. If B L, the e : Z is a Riesz basis i L, with frame bous B Ž L. a B Ž L., where B Ž L. is efie recursiely as follows: B Ž L. cos L si L, a for, B L B L B L B Ž L.. I, Chui a Shi improve this propositio. They prove the followig. PROPOSITION. Ž, Theorem.. The Ž uiue. zero of B Ž. t i 0, is a stability bou, where cos t si t B Ž t. Ž HŽ t.., HŽ t. si t HŽ t. cos t si t. t The stability bous obtaie i Propositio. a Propositio. are as follows :

3 STABILITY OF TRIGONOMETRIC SYSTEMS 6 TABLE I Stability Bous FavierZalik ChuiShi FavierZalik ChuiShi It is easy to see that both the FavierZalik bous a the ChuiShi bous are very small for large. I this paper, we show that is a stability bou for ay a that caot be improve. I fact, we prove the followig. THEOREM.. Let : Z be a seuece i R for which L sup ; Z the i², : e : Z forms a Riesz basis for L, with frame bous Ž. BŽ L. a Ž. BŽ L., where BŽ L. cos L si L. Moreoer, the ieuality i Ž. caot be replace by euality. i², : We call e : Z a Fourier frame, if it costitutes a frame for L,. For the case of oe imesio, Bala a Christese 3 gave the stability bous of Fourier frames. Here we give a multivariate versio. THEOREM.. Let Ž,...,. a Ž,...,. for i², : Z. If e : Z is a frame for L, with bous A a B a ' ' / AB L sup arcsi, ž Z i², : the e : Z is a frame for L, with bous A BA B L a B B L. ' The proofs of the previous two theorems as well as the proofs of FavierZalik a ChuiShi use the followig fact: if oe of the compoets of Ž,...,. chages, the oly the correspoig compoet i², : of chages. I other wors, these proofs are vali oly whe e : i i Z e e :,..., Z.

4 6 SUN AND ZHOU I geeral, whe oe of the compoets of Ž,...,. chages, all of the compoets of may chage. For this case, we have Ž. Ž. THEOREM.3. Let,..., a,...,, Z. i², : Suppose that e : Z is a frame for L, with bous A a B. If si L si L A DŽ L. cos L si L (, L L B 0 L a L, Z ; i², : the e : Z is a frame for L, with bous A ' BA D L a B D L. I particular, if, 0 L, DŽ L. a L, Z, i², : the e : Z is a Riesz basis for L, with bous Ž. DŽ L. a Ž. DŽ L.. Notatio. I this paper, the orms of all Hilbert spaces are eote by. The exact meaig ca be see by cotext.. PROOFS OF THE THEOREMS Proof of Theorem.. We will prove the theorem for. If, the proof is similar. i i By Kaec s -theorem, both e : Z a e : Z are Riesz bases for L, with bous BŽ L. a BŽ L.. Hece for ay fiite seuece of complex umbers c :, Z,we,

5 STABILITY OF TRIGONOMETRIC SYSTEMS 63 have c, e, iž. i i H H, c e e i H, B L c e i H, B L c e, B L c. A similar argumet shows that iž.,,, c e B L c. i i O the other ha, sice both e : Z a e : Z are iž. complete i L,, e :, Z is complete i iž. L,. By 7, Theorem.9 we kow that e :, Z is a Riesz basis for L, with bous Ž. BŽ L. a Ž. BŽ L.. Moreover, the couterexamples i 7, pp. 5 ca also be extee to the case of a multivariate, so the ieuality Ž. caot be replace by euality. I fact, a explicit couterexample was show i. To prove Theorem., we ee the followig lemma. i², : LEMMA.. Suppose,..., for Z. The e : i Z is a frame Riesz basis for L, if a oly if e k k : Z is also a frame Ž Riesz basis. for L, k for ay k. Moreoer, if the coitios are satisfie a A k a B k are the frame i bous for e k k : Z, the A A A A a B BB B. k Proof. Agai, we cosier the case. If, the proof is similar. First, we show the ecessity. Without loss of geerality, we take k.

6 6 SUN AND ZHOU i², : Let e : Z be a frame for L,. Fix some g L, such that g 0. For ay f L,, we have ² : iž., A fž. g Ž. f Ž. g Ž., e B fž. g Ž.. Sice f g f g, the above ieuality implies ² : ² : i i A fg f, e g, e B fg. ² i : Let D g g, e. The 0 D a ² : i AD f f, e BD f, f L,. i Hece e : Z is a frame for L,. For the case of Riesz basis, sice a Riesz basis is also a frame, we oly i ee to show that e is liearly iepeet. Fix some fiitely ozero complex seuece : Z such that 0. For ay fiite complex seuece c : Z, a similar argumet shows that i AD c c e BD c, i i where D e. This implies e is liearly iepeet i L,. i Next, we show the sufficiecy. Let e k k : Z k be a frame for L, with bous A k a B k, k,. For ay f L,,we have iž. f Ž,. e H H, Z i i H H Ž. f, e e i H H B f, e

7 STABILITY OF TRIGONOMETRIC SYSTEMS 65 i B H H fž,. e H H BB fž,.. Ž 3. A similar argumet shows that iž. f Ž,. e H H, Z H H A A fž,.. i², : Hece e : Z is a frame for L,. For the case of Riesz bases, the proof is similar to that of Theorem.. i k k Moreover, suppose that Ak a Bk are the frame bous for e : Z, k,. By Ž.Ž., 3, a Ž. k, it is easy to see that A AA a B BB. i k k Proof of Theorem.. For ay k, by Lemma., e : Z is a frame for L, k with bous Ak a Bk for which A AA A a B BB B. Sice BkAk, we have BkAk BA. Hece ' k A B k AB arcsi arcsi. ' ' i By, Theorem, e k k : Z is a frame for L, k with bous B Ak ' A B L a Bk B L. The coclusio follows by Lemma.. Proof of Theorem.3. Let k si Ž. si a0ž., akž., Ž k. k iž. cos ak Ž., k 0Ž x., kž x. cos kx, Ž. k x si k x, k. '

8 66 SUN AND ZHOU k k 0 k For ay k 0, a is icreasig for 0, a a L cos L si L Ž si L. L Žsee, 7.. Moreover, Hece k kz i x e a Ž. Ž x., x,. iž. e Ł kz a k Ž. k Ž., k Ž k,...,k.. For ay fiitely ozero seuece of complex umbers c, we have i², : i², : c Ž e e. Z i², : i², : c e Ž e. Z Ž Ž,...,.. c e a Ž. k ž Ł / i², : Ł k Ž. Z kz k c e a i², : 0 Z i², : Ł k Ł k k0 si L ' ž ž ž L / / / Ž. c e a Ž. B c ' Ł k Ž. B c a k0 P Q, Ž 5. where lettig G k Z : p compoets of k are 0 p, p we have: ' Ł k Ž. Q B c a p0 kg p

9 STABILITY OF TRIGONOMETRIC SYSTEMS 67 ' p Ł k p0 k p p,...,k0 a L B c ž / p ' p k p0 k0 ž ž L / / a L B c ž / si L cos L si L ' B c. Ž 6. Ž. Ž. i², : i², By 5 a 6, e e : is a Bessel seuece with bou Ž. i², : D L B. By, Theorem, e : Z is a frame for L, whe DŽ L. ' AB. This completes the proof. ACKNOWLEDGMENT The authors are grateful to the referees for their valuable suggestios. REFERENCES. R. Bala, Stability theorems for Fourier frames a wavelet Riesz bases, J. Fourier Aal. Appl. 3 Ž 997., O. Christese, A Paley-Wieer theorem for frames, Proc. Amer. Math. Soc. 3 Ž 995., O. Christese, Perturbatio of frames a applicatios to Gabor frames, i Gabor Aalysis a Algorithms: Theory a Applicatios ŽH. G. Feichtiger a T. Strohmer, Es.., pp. 9309, Birkhauser, Basel, C. K. Chui a X. L. Shi, O stability bous of perturbe multivariate trigoometric systems, Appl. Comp. Harm. Aal. 3 Ž 996., S. Favier a R. Zalik, O the stability of frames a Riesz bases, Appl. Comp. Harm. Aal. Ž 995., M. I. Kaec, The exact value of the Paley-Wieer costat, Soiet Math. Dokl. 5 Ž 96., R. Youg, A Itrouctio to No-Harmoic Fourier Series, Acaemic Press, New York, 980.