Ratio Mathematica Volume 41, 2021, pp. 197-205 On Cesàro’s Means of First Order of Wavelet Packet Series Manoj Kumar* Shyam Lal† Abstract Wavelet packets have the capability of partitioning the higher-frequency octaves to yield better frequency localisation. Ahmad and Kumar [2000] have obtained the pointwise convergence of the wavelet packet series. But till now no work seems to have been done to obtain Cesàro summability of order 1 of wavelet packet series. In an attempt to make an advanced study in this direction, a novel theory on (C, 1), Cesàro summability of order 1 of wavelet packet series is obtained in this study. Keywords: Multiresolution analysis, (C, 1) summability, wavelet pack- ets, periodic wavelet packets, wavelet packet expansions. 2020 AMS subject classifications: 40A30, 42C15. 1 1 Introduction Several researchers, including S. E. Kelly and Raphael [1994a], S. E. Kelly and Raphael [1994b], Kumar and Lal [2013], Meyer [1992], Walter [1992], Wal- ter [1995], Wickerhauser [1994], have investigated the problem of wavelet packet series convergence and demonstrated that wavelet packets are a basic yet effective wavelet and multiresolution analysis extension. Wavelet packet functions are a collection of functions that can be used to create other functions. Wavelet packet *Applied Sciences and Humanities Department, Institute of Engineering and Technology, Lucknow-226021, India; manojkumar@ietlucknow.ac.in. †Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi- 221005, India; shyam lal@rediffmail.com. 1Received on November 18, 2021. Accepted on December 12, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.687. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 197 Manoj Kumar, Shyam Lal functions are still time-localized, but they have more versatility in describing di- verse types of signals than wavelets. Wavelet packets, in particular, are better at encoding signals with periodic behaviour. Wavelet packets can partition higher- frequency octaves, resulting in more accurate frequency localization. Working in slight different directions, Ahmad and Kumar [2000] have ob- tained the pointwise convergence of the wavelet packet series. But till now no work seems to have been done to obtain Cesàro summability of order 1 of wavelet packet series. It is important to note that Cesàro summability is a strong tool to obtain the convergence than that of ordinary convergence. This work establishes a new theory on Cesàro summability of order 1 of wavelet packet series in an attempt to make a more advanced study in this field. 2 Definitions and Preliminaries Let L2(R) be the space of measurable and square integrable functions over set of real numbers R . If a function φ ∈ L2(R) generates nested sequences of closed subspaces, it is said to produce an MRA (multiresolution analysis), Qı = span{φı, : ı, ∈ Z}, where φı,(t) = 2ı/2φ(2ıt− ) and Z is the set of integers , satisfying the following conditions (i) ... ⊂ Q−2 ⊂ Q−1 ⊂ Q0 ⊂ Q1 ⊂ Q2 ⊂ ..., i.e. Qı ⊂ Qı+1, ı ∈ Z; (ii) (∪ı∈ZQı) = L2(R); (iii) ∩ı∈ZQı = {0}; (iv) λ(t) ∈ Qı ⇔ λ(2t) ∈ Qı+1, ı ∈ Z such that φ0, form a Riesz basis of {Q0}. A function φ which generates a multires- olution analysis, is called a scaling function . Wavelet packets can be constructed with the help of multiresolution analysis. We know that if H is a Hilbert space with ONB (orthonormal basis) {�}∈Z then, λ2k = √ 2 ∑ ∈Z α2k−�, λ2k+1 = √ 2 ∑ ∈Z β2k−�, where {αk}k∈Z and {βk}k∈Z are in l2(Z), are orthonormal bases of two orthogonal closed subspaces H1 and H0 respectively, such that H = H1 ⊕H0. Using the foregoing decomposition strategy, we now build the fundamental wavelet packets connected with the scaling function φ ∈ L2(R) which is already defined in multiresolution analysis. 198 On Cesàro’s means of first order of wavelet packet series Let {ξk, k = 0, 1, 2, ...,} denote a wavelet packet family that corresponds to the scaling function φ which is orthonormal. Consider ξ0 = φ. Recursively, the wavelet packets ξk, k = 0, 1, 2, ..., are defined by  ξ2k(t) = √ 2 ∑ ∈Z hξk(2t− ) ξ2k+1(t) = √ 2 ∑ ∈Z gξk(2t− ). (1) As a result, the {ξk} family is a generalisation of the orthonormal wavelet ξ1 = ψ, often known as the mother wavelet. For the Hilbert space L2(R), the set {ξk(t−) : k = 0, 1, 2, ...,  ∈ Z} form an ONB. Consider the family of subspaces of L2(R) as Pkı = span{2 ıξk(2 ıt− ) :  ∈ Z}, ı ∈ Z, (2) formed by the family of wavelet packets {ξk} for each k = 0, 1, 2, .... Observe that P0ı = Qı and P 1 ı = Wı, where {Qı} is the multiresolution analysis of L2(R) produced by ξ0 = φ and {Wı} is the sequence of orthogo- nal complimentary subspaces generated by the wavelet ξ1 = ψ. The orthogonal decomposition Qı+1 = Qı ⊕Wı, ı ∈ Z can then be expressed as P0ı+1 = P 0 ı ⊕P 1 ı . (3) As follows, this orthogonal decomposition can be extended from k = 0 to any k = 1, 2, 3, ... in the form of Pkı+1 = P 2k ı ⊕P 2k+1 ı , ı ∈ Z. (4) Now we’ll state a result that will be employed in the theorem’s proof. The decomposition trick (4) produces Wı = P 1 ı = P 2 ı−1 ⊕P 3 ı−1 = P4ı−2 ⊕P 5 ı−2 ⊕P 6 ı−2 ⊕P 7 ı−2 ... = P2  ı− ⊕P 2+1 ı− ⊕ ...⊕P 2+1−1 ı− ... = P2 ı 0 ⊕P 2ı+1 0 ⊕ ...⊕P 2ı+1−1 0 , (5) for each ı = 1, 2, ..., where (2) declares Pkı . Furthermore, the family { 2 ı− 2 ξr(2 ı−t− l) : l ∈ Z } is an ONB of Prı−, where r = 2  + µ for each µ = 0, 1, 2, ..., 2 − 1,  = 1, 2, ...ı; 199 Manoj Kumar, Shyam Lal and ı = 1, 2, .... All of the elements of this base, however, have the same basic shape: ξı,k,(t) = 2 ı/2ξk(2 ıt− ). (6) Let λ ∈ L2(R), then the function λ can be approximated by a wavelet packet series as follows: λ(t) ∼ ∑ ı∈Z 2r+1−1∑ k=2r ∑ ∈Z Cl,k,ξl,k,(t), (7) where l = ı− r, r = 0 if ı < 0 and r = 0, 1, 2, ..., ı if ı ≥ 0; and the coefficients Cl,k, defined by Cl,k, = 〈λ,ξl,k,〉 , (8) are called the wavelet packet coefficients. Wavelet packets are a scalable time signal analysis method that combines the advantages of windowed Harmonic and wavelet processing. Wavelet bundles, which are periodic as well, offer a fascinating supplement to Fourier series. Using the periodization techniques for period 1 on the basis functions, an MRA for L2(R) can be transformed into an MRA for L2(0, 1). Let {ξk : k ∈ Z} denote the family of wavelet packets presented previously which is nonstationary in nature. Define general periodic wavelet packets ξperk,ı, by ξ per k,ı, = ∑ l∈Z 2ı/2ξk(2 ı(t + l) − ) for 0 ≤  < 2ı and k,ı = 1, 2, 3, · · · . With ξperk , We now define an operator Sνλ as follows: (Sνλ)(t) = 2r+1−1∑ k=2r ν∑ =0 〈 λ,ξ per l,k, 〉 ξ per l,k,(t). (9) Let sk = k∑ ν=0 aν be the kth partial sum of an infinite series ∞∑ k=0 ak. If σk = 1 k+1 k∑ ν=0 sν → s as k →∞ then the series ∞∑ k=0 ak is called summable to s by (C, 1) i.e. Cesàro means of order 1 (Titchmarsh Titchmarsh [1939]). Let Dµ(µ = 1, 2, 3, · · ·) be the collection of constant dyadic step functions on the intervals [2−µ, (+ 1)2−µ); 0 <  ≤ 2µ. Let D = ∪∞µ=1Dµ. Let B be a Banach space and σζ be a bounded linear functional on B which must be generated by any function ζ ∈ D as σζλ = ∫ 1 0 λζ for λ ∈ B. 200 On Cesàro’s means of first order of wavelet packet series We have |σζλ| ≤ ‖ζ‖∞‖λ‖B . Now if we take B = Lq and define ‖ζ‖r = ‖σζ‖ = sup ‖λ‖q≤1 ∫ 1 0 λζ for any ζ ∈ D. (10) Then clearly ∣∣∣∣ ∫ 1 0 λζ ∣∣∣∣ ≤‖λ‖q ‖ζ‖r ,λ ∈ Lq,ζ ∈ D. (11) Let us write Πıλ(t) = 2ı−1∑ µ=0 ( 1 µ + 1 µ∑ ν=0 (Sνλ)(t) ) δ[µ2−ı,(µ+1)2−ı) = 2ı−1∑ µ=0 σµλ(t)δ[µ2−ı,(µ+1)2−ı) and Aı = 2ı−1∑ µ=0 C per l,k,δ[µ2−ı,(µ+1)2−ı), where (9) defines Sνλ and δI is the characteristic function on I ⊂ R. We’re going to define an operator now Tı(t,x) = 2 −ı 2ı−1∑ =0 C per l,o,φ per ı, (t)φ per ı, (x) = 2−ı 2r+1−1∑ k=2r ∑ µ<ı 2ı−1∑ =0 ξ per l,k,(t)ξ per l,k,(x), where l = µ− r, r = 0 if µ < 0 and r = 0, 1, 2, ...,µ if 0 ≤ µ < ı. In this paper, an estimate for the Cesàro summability of wavelet packet series has been determined in the following form: Theorem 2.1. Let λ be 1-periodic continuous function. Then∥∥∥∥∥∥ ( 2−ı 2ı−1∑ µ=0 ∣∣∣∣∣ 1µ + 1 µ∑ ν=0 Sνλ ∣∣∣∣∣ r)1/r∥∥∥∥∥∥ ∞ ≤ C‖λ‖∞ (12) if and only if ‖Tı‖1 ≤ C‖Aı‖q , (13) 201 Manoj Kumar, Shyam Lal where C > 0, a constant and 1 < r < ∞. Furthermore, lim ı→∞ ‖Πıλ(t) −λ(t)‖r = 0 uniformly in [0, 1]. Proof. By equation 12 we have ( 2−ı 2ı−1∑ µ=0 ∣∣∣∣∣ 1µ + 1 µ∑ ν=0 Sνλ ∣∣∣∣∣ r)1r = ‖Πıλ (t)‖r = sup ‖Aı‖q≤1 2−ı 2ı−1∑ µ=0 C per l,k,σµλ (t) = sup ‖Aı‖q≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 C per l,k,Sνλ (t) = sup ‖Aı‖q≤1 ∫ 1 0 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 C per l,k,Kν (t,x) λ(x)dx ≤ ‖λ‖∞ sup ‖Aı‖q≤1 1 µ + 1 µ∑ ν=0 ‖Tν(t,x)‖1 ≤ ‖λ‖∞ sup ‖Aı‖q≤1 1 µ + 1 µ∑ ν=0 ( C‖Aı‖q ) , by (13) = ‖λ‖∞ sup ‖Aı‖q≤1 C‖Aı‖q ≤ C‖λ‖∞ , where Kı (t,x) = 2ı−1∑ =0 φperı, (t)φ per ı, (x) = 2r+1−1∑ k=2r ∑ µ<ı 2ı−1∑ =0 ξ per l,k,(t)ξ per l,k,(x). If, on the other hand, (12) is true, we have 202 On Cesàro’s means of first order of wavelet packet series ‖Tı(t,x)‖1 = sup ‖λ‖∞≤1 ∫ 1 0 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 C per l,k,Tν(0,x)λ(x)dx = sup ‖λ‖∞≤1 ∫ 1 0 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 C per l,k,(2 −ν 2r+1−1∑ k=2r 2ν−1∑ =0 ξ per l,k,(0)ξ per l,k,(x))λ(x)dx = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 C per l,k,(2 −ν 2r+1−1∑ k=2r 2ν−1∑ =0 ξ per l,k,(0)) ∫ 1 0 λ(x)ξ per l,k,(x)dx = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 C per l,k,(2 −ν 2r+1−1∑ k=2r 2ν−1∑ =0 〈 λ,ξ per l,k, 〉 ξ per l,k,(0)) = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 C per l,k,(Sνλ)(0) = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 C per l,k,(σµλ)(0) = sup ‖λ‖∞≤1 ∫ 1 0 Πıλ(0)Aı ≤ sup ‖λ‖∞≤1 ‖Aı‖q ‖Πıλ(0)‖r ≤ ‖Aı‖q sup ‖λ‖∞≤1 ∥∥∥∥∥∥ ( 2−ı 2ı−1∑ µ=0 ∣∣∣∣∣ 1µ + 1 µ∑ ν=0 (Sνλ)(0) ∣∣∣∣∣ r)1/r∥∥∥∥∥∥ ∞ = ‖Aı‖q sup ‖λ‖∞≤1 ∥∥∥∥∥∥ ( 2−ı 2ı−1∑ µ=0 |(σµλ)(0)| r )1/r∥∥∥∥∥∥ ∞ , by (12) ≤ ‖Aı‖q sup ‖λ‖∞≤1 C‖λ‖∞ ≤ C‖Aı‖q . Now Πlλ(t) −λ(t) = M∑ µ=0 ((σµλ)(t) −λ(t)) δ[µ2−l,(µ+1)2−l) = M∑ µ=0 1 µ + 1 µ∑ ν=0 ((Sνλ)(t) −λ(t)) δ[µ2−l,(µ+1)2−l) 203 Manoj Kumar, Shyam Lal for any l ≥ M ≥ 2ı. As a result, ‖Πlλ(t) −λ(t)‖ ≤ M∑ µ=0 ∥∥∥∥∥ 1µ + 1 µ∑ ν=0 ((Sνλ)(t) −λ(t)) ∥∥∥∥∥ ∞ ∥∥∥δ[0,2−l)∥∥∥r ≤ M∑ µ=0 1 µ + 1 µ∑ ν=0 ‖Sνλ−λ‖∞ ∥∥∥δ[0,2−l)∥∥∥r , since the limit of the characteristic function of [0, 2−ı) in all Lr-space (1 < r < ∞) is 0 and thus the ultimate result is fallowed. The theorem’s proof is now complete.2 3 Conclusions The estimate for the Cesàro summability of order 1 of wavelet packet series has been determined in the form of lim ı→∞ ‖Πıλ(t) −λ(t)‖r = 0 uniformly in [0, 1]. Acknowledgements. One of the authors, Shyam Lal, is grateful to DST-CIMS for supporting his work. Manoj Kumar is thankful to the Director, Institute of Engineering and Tech- nology, Luchnow for promoting this research activity. References K. Ahmad and R. Kumar. Pointwise convergence of wavelet packet series. Atti Sem. Fis. Univ. 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