On the Convergence of Jacobi Series at the Poles

Transcription

1 On the Convergence of Jacobi Series at the Poles *1 Sarjoo Prasad Yadav, 2 Rakesh Kumar Yadav, 3 Dinesh Kumar Yadav 1 Dept of Maths, Govt. Science College, Rewa, M. P. India; 2 RBY PG College, Gulalpur, Purvanchal University, Jaunpur, U.P. India; 3 Govt. Polytechnic, Hardoi, U.P. India *1 ydsrpa@yahoo.co.in, 3 yadavdinesh25@gmail.com Abstract The most urgent need in inmation Technology is data compilation different purposes in varying ways. Convergence of Jacobi series is used replacement of discontinuous signals by its approximated absolutely continuous signals data manipulation in computers. Here in this paper a Banach space X of signals which are p-power Lebesgue integrable with weight on [-1,1] is considered. Some of subspaces of X have been recognized by the convergence behavior of Fourier-Jacobi expansions associated with the signals. These results are applied to signal processing with wavelets related to useful concept in Science Engineering disciplines. (1.1) is nth Fourier-Jacobi Transm of such that (1.2) (1.3) is orthonormalized Jacobi polynomial, 1,2,.) (1.4) Keywords Fourior-Jacobi Expansion; Signal Processing; Data Repairs; Wavelets (1.5) Introduction (1.6) Convergency of an infinite series has always been a challenge to mathematicians. Convergence of Jacobi series not only leads to unim convergence or an approximation over the interval [-1, 1] but explains transm of a signal into absolutely continuous signal. On on this has potential applications such as contaminated noise removal, corrupted data repairs many more in inmation technology in the Science Engineering branches. Let X denote either the space C[-1,1] of all continuous signals or the space of p-power Lebesgue integrable signals with weight on [-1,1]. Sup p-norms are defined as usual. A series called Fourier-Jacobi expansion is associated (see Szegö [3], Chapter IX) with every as is the nth Jacobi polynomial of order degree (see Szegö [3] ). is the Kronecker delta. To avoid confusions have easy access to verifications of mulae, the notations of Szego are employed [3]. As consequence, we write (1.7) (1.8) Hence the study of Jacobi series will make no 19

2 Frontiers in Environmental Engineering (FIEE) Volume 2 Issue 2, June 2013 confusion when one uses orthonormalized Jacobi polynomials in place of, divided by. Convergency of Jacobi Series at End Points It is obvious that every signal of the space X can be associated to a unique Jacobi series given by (1.1) if the integrals in (1.7) exist every Unique is in the sense that two signals are treated the same if they are equal almost every in the interval of their definitions. It is natural to examine the circumstances in which this series (1.1) of Jacobi polynomials converges to the strength of the signal of which it is an outcome. The first result in this direction, known to us is that of Raü [2] continuity of the signal in the whole interval [-1, 1] is required is must. Improving the results of (Szegö [3] Chapter IX theorem 9.1.4), Yadav, S. P. [4] has recognized the subspaces, weaker than the continuity of extra at the other end exclude from the case is required at the pole something. Even then these spaces recognized in Yadav [4]. Moreover, Szegö ([3]( ) page 262 see the case k = 0) shows that continuous functions/signals on [-1,1] exist in X so that its associated Jacobi series diverges on. It is natural to find the circumstances in which the case is covered. It is shown that the Jacobi series associated to signals which satisfy an integrability pole condition (2.1), converges to A at. We assume as t +0, A is a constant depending on (2.1) only. This is a condition on at. The condition (2.1) is independent of the continuity of in [-1, 1] or at but stronger than that assumed in Yadav [4]. The condition in Yadav [4] at is (2.2) = (2.3) by (2.1) as t +0. However, its converse is obviously not always possible. On the other h continuity implies (2.2) but not (2.1). As indicates that the integral in (2.1) diverges while (2.2) holds. Moreover, the condition (2.1) is not trivial as functions of class satisfy it. Let A = then (2.4) as +0, are absolute constants, but not the same every in this article until unless stated otherwise. This shows that the class Lip satisfies (2.1). Lipschitz condition on is necessary only in the arbitrarily small neighborhood of Moreover, signals are expected to satisfy the condition (2.1). The conditions on are called antipole conditions e assumed as follows just the same as in Yadav [4]. (2.5) as +0, In certain cases a condition lighter than (2.5) is assumed as (2.6) as +0,. Following end point convergency results of Jacobi series are mile stones in the literature of Jacobi series as some results can be found on representing the signals in terms of Jacobi polynomials wavelets. The pattern of notations is adopted in continuation of Yadav [4] the followings are proved: Theorem 1 Let satisfy the pole condition (2.1). Then the Jacobi series (1.1) associated to converges to A at the point. as t +0, A is a constant depending on For all, (2.1) is implied (2.2) as only. Theorem 2 Let satisfy the pole condition (2.1) 20

3 but. Then the Jacobi series (1.1) associated to converges to A at the point. To overcome the restriction of lighter antipole condition (2.6) so that, we have. Theorem 3 Let, we need a satisfy the pole condition (2.1) along with the antipole condition (2.6) associated to but. Then the Jacobi series (1.1) converges to A at the point. Improving the heaviness of the antipole condition leads to the removal of the restriction from to get the following: Theorem 4 Let satisfy the pole condition (2.1) along with the antipole condition (2.5). Then the Jacobi series (1.1) associated to converges to A at the point. All these spaces are normalized Banach subspaces as shown in Yadav [4]. Our theorems 1, 2, 3 4 hold good at the other end point, only change of is apparent the antipole condition should hold at. Lemmas Known Results to be Used Following order estimates of the Jacobi polynomials are taken from Szegö [3] to prove our theorems. Lemma 3.1 (Szegö [3] theorem (7.32.2). Let arbitrary real c a fixed positive constant,. Then be Lemma 3.3. (Szegö [3] page 71 (4.5.3)) For arbitrary we have Lemma 3.4 (Askey Wainger [1] page 470 ) For arbitrary we have any λ > 0 absolute contants. Lemma 3.5 Let a signal (3.3) (3.4) satisfy as in the respective spaces with the required antipole condition. Then the following estimate. holds as, Proof: We break the integral in (3.5) as (3.5) (3.6) Moreover, (3.1) c, are arbitrarily small but fixed positive reals. The second order of Jacobi polynomial given arbitrary in (3.1) is use to get. (see Szegö [3] page respectively). Lemma 3.2 (Szegö [3] theorems (8.21.8) ( ). Let be arbitrary real numbers,. Then by the pole condition (2.1) by the first equation of (3.1), we have (3.7). Again (3.2) 21

4 Frontiers in Environmental Engineering (FIEE) Volume 2 Issue 2, June 2013 But, let us write examine the meaning of the condition (2.1) which tells that Theree, by the antipole condition (2.6) when chosen arbirarily small positive real. Moreover, (3.12) arbitrary. (3.8) Now, to calculate we use the first order given in (3.2) which holds unimally in but fixed then apply the Riemann-Lebesgue lemma i.e. (3.13) by condition (2.6) as. Similarly, in Theorem 4 we have condition (2.5) so that (3.14) as. Writing (3.9) To counter the orders of we run through the situations given in Theorems 1, 2, 3 4. According to Theorem 1, we have only. Thus using the estimate of from (3.1), we get so that by the meaning of the condition (2.5). Hence Choosing arbitraily small, (3.10). (3.15) as in (3.11) of (3.10) by using the orders of Jacobi polynomials valid in. In the situation of Theorem 2, we have exactly the same calculations, only keeping in mind that in case we have to suppose. In Theorems 3 4 we use the antipole conditions (2.6) (2.5) respectively. Thus Theorem 3, by the condition (2.5), noting that so that. This completes the proof of the Lemma 3.5. Proof of the Theorems 1, 2, 3 4. Let denote the nth partial sum of the Jacobi series at then, 22

5 Thus any absolute constant A, by the orthogonality of Jacobi polynomials, we have by Lemmas So. (4.1) (4.2) by Lemma 3.5. This completes the proof Theorems 1, 2, 3 4. Conclusion In conclusion, it can be seen that signals having countable number of discontinuities of first kind are associated with a Jacobi series which is convergent at the end points called Poles. This indicates that a discontinuous signal can be approximated by unimly convergent Jacobi series waveletsso that a contaminated voice signal or corrupted data can be repaired with due techniques many more other applications. REFERENCES Askey, R. Wainger, S. A Convolution Structure Jacobi Series, Amer. J. Math XCI (1969) MR 41# Raü H. Uber die Lebesgueschen constantan der Reihenentwicklungen nach Jacobischen Polynomen, Journal für die reine und angewte Mathematik 161 (1929) Szegö, G. Orthogonal Polynomials. AMS Collo. Publication 3rd ed. XXIII (1967). New York. MR 46# Yadav, Sarjoo Prasad. On the Characterization of Function Spaces in Terms of Sequential Properties, J. Indian Math Soc. 69 (2002) MR 2004m # Corresponding address: Plot no. 74, Sarai Taki, Chhatnag Road, Jhunsi, Allahabad. U.P. India. ydsrpa@yahoo.co.in Dr Sarjoo Prasad Yadav, Ph.D is exfaculty of Mathematics in Science College at APS University, Rewa (M.P) INDIA. He is working as senior Scientist. He has written 51 research papers a graduate course book on Analysis, in addition to that, he has supervised five Ph.D s. He is life member of many Indian Learned Societies including American Math. Soc. reviewer of Mathematical Reviews since Corresponding author, his mailing address: Jay Ganga Maiya Bhawan. Polt No. 74, Sarai Taki. Chhatnag Road.JHUNSI. ALLAHABAD, (U. P.) India. ydsrpa@ yahoo.co.in Rakesh Kumar Yadav, M. Sc. is a research scholar of APS University Rewa (M.P) INDIA. He is presently at the Faculty of RBY P-G College, Gulalpur, Jaunpur, U.P. India. Dinesh Kumar Yadav, M.Tech. is lecturer in Govt. Polytechnic,Hardoi (U.P). Formerly he was on the faculty of JSS Academy of Technical Edn. Noida (U.P) India. yadavdinesh25@ gmail.com 23