Fractional Hankel and Bessel wavelet transforms of almost periodic signals

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1 ÜnalmışUzunJournal of Inequalities and Applications 15) 15:388 DOI /s R E S E A R C H Open Access Fractional Hankel and Bessel wavelet transforms of almost periodic signals Banu ÜnalmışUzun * * Correspondence: buzun@isikun.edu.tr Department of Mathematics, Işık University, Şile,Istanbul, 3498, Turkey Abstract The main objective of this paper is to study the Hankel, fractional Hankel, and Bessel wavelet transforms using the Parseval relation. We construct a generalized frame and write new relations and inequalities using almost periodic functions, strong limit power signals, and these transform methods. MSC: 4A75; 4A4; 4A38 Keywords: almost periodic function; fractional Hankel transform; Bessel wavelet transform; frame 1 Introduction The Hankel transform HT) is a well-known integral transform which uses the th order Bessel function of the first kind, J, as a kernel. Since the HT is equivalent to the twodimensional Fourier transform FT) of a circularly symmetric function, it plays an important role in a number of applications including optical data processing, digital filtering, etc. [1 3]. The conventional HT is extended to the fractional Hankel transform FrHT) by Kerr [4] and Namias [5]. Its propertiesarediscussed in detail by several authors [6 9] and its applications in many areas such as optics, signal processing, quantum mechanics) are given in [1 1]. Using the theory of Hankel translation, Pathak and Dixit defined continuous and discrete Bessel wavelet transforms BWT) and studied their properties [13].The fractional Hankel transformation and the continuous fractional Bessel wavelet transformation, some of their basic properties and applications are studied in [14]. In [15], the relation between the Bessel wavelet transformation and the Hankel-Hausdorff operator is discussed. In this paper, we first of all introduce the FrHT, the BWT, and almost periodic functions and some of their properties in brief. A generalized frame decomposition for almost periodic functions is constructed by using an orthogonal basis with Laguerre functions relating to the FrHT. We also give various relations using the HT, FrHT, BWT, and strong limit power signals. 1.1 Hankel and fractional Hankel transforms We define the Fourier transform of an integrable function f as ˆf t)ff)t) 1 f x)e itx dx. π 15 Ünalmış Uzun. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page of 1 The general HT is given as H [f ]x) yj yx)f y) dy, Re)> 1 1.1) and the inverse is given by f y) xh [f ]x)j yx) dx, where J is the th order Bessel function of the first kind [16]. The HT of order zero is an integral transform equivalent to a two-dimensional FT with a symmetric integral kernel, H [f ]q) 1 π 1 π 1 π rj qr)f r) dr 1 π π π φ φ π ) e irq cos θ dθ rf r) dr e irq cos θ rf r) dr dθ e irq cosθ φ) rf r) dr dθ f x, y)e iux+vy) dxdy ˆf u, v), 1.) where r x + y and q u + v.thisisalsoknownasthefourier-besseltransform. Also Parseval s theorem holds for the HT: xh [f ]x)h [g]x) dx yf y)gy) dy. Namias [5] introduced the concept of Fourier transform and Hankel transform of fractional order FrFT and FrHT), opening the new period of fractional transforms. FrFT with angle α of a signal f x)isgivenas F α y) ei π 4 α ) e i x +y cot α+ixycsc α f x) dx 1.3) π sin α and the inverse is f x) e i π 4 α ) π sin α e i x +y The Parseval relation for FrFT is given as f x) dx cot α ixycsc α F α y) dy. 1.4) F α y) dy. 1.5)

3 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 3 of 1 The FrHT is defined by where with H α [f ]y) f x)k α x, y) dx, 1.6) ) K α x, y)a,α xe i x +y ) cot α xy J sin α A,α ei+1) π α ) sin α. The inverse is given by where f x) as in [17]. H α [f ]y)k αx, y) dy, 1.7) K α x, y) e i+1) π α ) ) sin α xe i x +y ) cot α xy J sin α 1. Orthogonal basis with Laguerre functions Let L n be the generalized Laguerre polynomials defined by means of the Rodrigues formula L n x) 1 n! ex x D n[ e x x n+], n N N {}, x > which gives L n x) 1) k n + )! n k)! + k)!k! xk k in power series form. The Laguerre functions are defined as ln x)1 [, ]x)e x/ x / L n x) and they form an orthogonal basis for the space L, ). Let us set FSn x) l n x) and take a) n aa +1)a +)...a + n 1). It was shown that S n x) Ɣ + 1)1 + ) n n! n) k +1) k 1 1 k! +1) k ix k ) k+ +1 and the Sn are orthogonal on the real line [18].

4 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 4 of 1 The functions Sn can be written as a linear combination of Paul s wavelets, such that ) 1 α+1 ψ α x), x + i S n ) x C c k ψ k+ x), 1.8) k where the constants C and the coefficients c k defined as and C Ɣ + 1)1 + ) n n! c k i) k+ +1 n) k +1) k k! +1) k. Sn;a,b x)a/ Sn a x b)isthenaturaldiscretizationofsn x)foreverya, b Z. If another complete orthogonal basis {ψn } n,givenby ψ n x)x e x L n x ), is chosen, we get H [ y e y L n y )] x)e inπ x e x L n x ), n N, x >, using equation 7.414) in [19]. Then the integral representation ) H α [f ]x)a,α ye i x +y ) cot α xy J sin α f y) dy, 1.9) with A,α ei+1) π α ) sin α, whichisknownasthefrhtisobtained[5]. 1.3 Continuous Bessel wavelet transform Let σ x) γ +1 x γ +1/ Ɣγ + 3 ) 1.1) and jx)c γ x 1/ γ J γ 1/ x), C γ γ 1/ Ɣ γ + 1 ), 1.11)

5 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 5 of 1 where γ is a positive real number and J γ 1/ x) is the Bessel function of the first kind of order γ 1.Define Dx, y, z) jxt)jyt)jzt) dσ t) [Ɣ 3γ 5/ γ + )] 1 [ Ɣγ )π 1/ ] 1 xyz) γ +1 [ x, y, z) ] γ, where x, y, z) denotes the area of a triangle with sides x, y, z if such a triangle exists and is zero otherwise. x, y, z) is nonnegative and symmetric in x, y, z. L p σ, ), 1 p < is the space of measurable functions φ on, ), such that [ ] 1/p φ p,σ φx) p dσ x) <, 1 p <, φ,σ ess sup <x< φx) <. Let ψ L p σ, ), 1 p < be given. The Bessel wavelet is given by ψ b,a x):a γ 1 Db/a, x/a, z)ψz) dσ z), 1.1) where a >andb. If ψ L σ, ), it satisfies the admissibility condition see [15]) C μ,ψ x μ H [ψ]x) dx <, μ 1, 1.13) where H [ψ]x) is the Hankel transform of ψt). Using the Bessel wavelet, Pathak and Dixit [13] introduced the continuous Bessel wavelet transform BWT) as B ψ f )b, a) f, ψ b,a t) a γ 1 Also Pathak et al. []statedtheequality f t)ψ b,a t) dσ t) b f t)ψz)d a, t ) a, z dσ z) dσ t). ) b B ψ f )b, a)f φ), a >, 1.14) a using the convolution operator. 1.4 Almost periodic functions The space AP of almost periodic functions is the closure of quasi-periodic functions in the space L p loc of f,where f p is locally Lebesgue integrable on R for p 1. This space is definedastheclosedsubspaceofl R) given as the closed linear span of all functions e iλt where λ R see [1, ]). Equivalently, it is the completion of the space of trigonometric

6 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 6 of 1 polynomials on R whose elements can be written as n k1 a ke iλ kt,wheren N, a k C,and λ k R.AllAP functions are uniformly continuous and bounded, and we have f AP lim 1 T T T T f t) dt. Let QR) consist offunctions q in the form { m l1 qt) λ lt α l, t, m l1 λ l t) α l, t <, 1.15) where m 1,,...,λ l R, l 1,,...,m,andα 1 > α > > α m >. A function of the form Pt) a k e iq kt) k1 is called a generalized trigonometric polynomial on R, wherea k C, q k t) QR), and k 1,,...,n.DenotebyGtrigR) the set of all such polynomials. Afunctionf on R is said to have a strong limit power if for every ε >thereexistsa P ε GtrigR), such that f P ε sup { f t) P ε t) : t R } < ε. 1.16) Denote by SLPR) the set of all such functions. It is obvious that APR) SLPR). The inner product of the SLPR)spaceis defined by 1 f, g : lim T T see [3, 4]). T T f t)ḡt) dt Main results Proposition 1 Let f AP. Then lim T 1 T T π T q H [f ]q) dq dφ dt f L x,y),apt)..1) Proof Using the equality 1.) and the Parseval theorem for the two-dimensional Fourier transform, we get π q H [f ]q) dq dφ ˆf u, v) du dv f x, y) dxdy and the result follows. Lemma 1 Let F α y) be in L 1, ) for > 1.Then H α [ e x f x) ] C α gy)f α y) dy,.)

7 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 7 of 1 where C α ei[ π α )+ π 4 α 4 ] sin α + 3 π and gy) 1+iy csc α ) 3 1+iy csc α ) + 1+iy csc α ) 1 ) ) y sin α 1 + iy csc α )1 + iy csc α ). 1 Moreover, if gy) L, then the FrHT satisfies H α [ e x f x) ] C α g f..3) Proof Using the definition of the FrHT 1.6)andtheinverseFrFT1.4), we get H α [ e x f x) ] ) A,α xe x e i x +y cot α xy J sin α e i π 4 α 4 ) e i x +y cot α ixycsc α F α π sin α y) dy dx. Changing the order of integration and using Theorem.1 in [5], H α [ e x f x) ] Therefore, A,α e i π 4 α 4 ) π sin α A,α e i π 4 α 4 ) π sin α 1+iy csc α ) F α y) e 1+iy csc α xy )x xj sin α dxdy F α y) 1+iy csc α ) 3 ) + 1+iy csc α ) 1 ) y sin α 1 + iy csc α )1 + iy csc α ) 1 ei[ π α )+ π 4 α 4 ] sin α + 3 gy)f α y) dy. π ) dy [ H α e x f x) ] C α gy) F α y) dy C α g F α y) dy C α g f by using the Parseval relation 1.5).

8 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 8 of 1 Theorem 1 Let f be an almost periodic function and let α be an angle where cot α >and α kπ, k is an integer. Then the FrHT of f is a strong limit power function in y if <x < λ k cot α. Proof Let f x) n k1 a ke iλkx be a trigonometric polynomial where < x < λ k cot α.then H α [f ]y)ei+1) π α ) sin α A,α a k k1 A,α a k k1 n A,α xe i x +y ) cot α k1 xe i x +y ) cot α +iλ kx 1) n y +n n! + n)! sin α x +n+1 e i x cot α +iλ kx dx a k k1 n ) xy a k e iλkx J sin α dx 1) n n i e y cot α )+n 1) n y +n n! + n)! sin α e )+n x +n+1 e i cot α x λ k cot α ) dx. xy sin α ) +n n! + n)! i y cot α + i λ k cot α Using the substitution u x λ k cot α,wewrite x +n+1 e i cot α x λ k cot α ) dx u + λ k cot α +n +1 l l ) +n+1 e i cot α u du ) u l λk cot α ) +n+1 l e i cot α u du.4) since the binomial series converges for < x < λ k cot α. This enables us to use Fubini s theorem and we have x +n+1 e i cot α x λ k cot α ) dx +n +1 l l ) λk ) +n+1 l cot α u l e i cot α u du..5) By [6], we know that 1 l+1 Ɣ )/ i cot α ) l+1, l > 1, u l e i cot α u m 1)!! du i m+1 cot α ) m π i cot α, l m, m integer, m!/ i cot α )m+1, l m +1,m integer,

9 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 9 of 1 since cot α >, we find a constant C m,nα, λ k ) as the solution of.4). So H α [f ]y) ei+1) π α ) sin α )+1 1) n n! + n)! sin α y+n )n n a k C m,n α, λ k )e i y cot α + i k1 λ k cot α, which shows that it is a generalized trigonometric polynomial in y. If f is a general almost periodic function, then there exists a sequence f n ) of trigonometric polynomials where f n f uniformly. Since H,n α y) GtrigR), we get Hα y) SLPR). Thus it is sufficient to verify that, if f n f, then H,n α y) Hα y) L y). Using the definition of the FrHT, it is easy to see that and H α y) f x) x J dx H,n α y) Hα y) fn x) fx) x J dx when < x < λ k cot α.since H,n α y) Hα y) L y) sup H,n α y) Hα y), the result follows. Theorem Let f be an almost periodic function and Sn;a,b x)a/ Sn a x b) be an orthogonal basis. Then there exist constants A, B >, such that A f AP a Z Proof We have f, S n;a,b a/ C lim N 1 N +1 N f, Sn;a,b B f AP..6) b N c k f x)ψ k+ a+1 x b ) dx k C c k f, ψ k+ ;a+1,b k C c k W ψk+ f )b, a +1), k where W ψ f )b, a) f, ψ a,b shows the wavelet transform of f.sincef is an almost periodic function, W ψk+ f is an almost periodic function in b as well [7]. Therefore

10 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 1 of 1 { f, Sn;a,b } b is a sequence of almost periodic functions. If f is a trigonometric polynomial f x) K l1 a le iλlx,weget f, S n;a,b a/ C Therefore, c k k a +1) C K l1 c k k a l ψ k+ a+1 x b ) e iλlx dx K l1 ) a l e iλ l b a+1 ˆψ λl k+. a+1 lim N 1 N +1 C [ k + N i k,k i b N a f, Sn;a,b [ K c k a l h k λ l )+ ] a m ā l j k λ l, λ m ) l1 [ K c i c k a l h i,k λ l )+ ]] a m ā l j i,k λ l, λ m ),.7) l1 where the sum is taken over those l, m such that λ m λ l is a nonzero) multiple of with π a+ b and h k λ l ) a j k λ l, λ m ) h i,k λ l ) a+) λl ˆψ k+, a+1 a a j i,k λ l, λ m ) ) ) a+) λl λm ˆψ k+ ˆψ a+1 k+, a+1 ) a+) λl ˆψ i+ a+1 a In this case, a m ā l j k λ l, λ m ) a λ ā λ+ π a+ s b λ R s Z\{} s Z\{} a λ R a λ R ) λl ˆψ k+, a+1 ) ) a+) λm λl ˆψ i+ ˆψ a+1 k+. a+1 a a λ ˆψ k+ λ a+1 λ a λ+ π a+ s ˆψ k+ b a+1 ) λ a+) ˆψ k+ a+1 ˆψ k+ ˆψ k+ ) λ πs ˆψ k+ + a+1 b λ πs + a+1 b ) 1/ λ πs + a+1 b ) 1/

11 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 11 of 1 s Z\{} a λ R a λ R a λ ˆψ k+ λ a+1 λ a λ πs ˆψ k+ a+1 b a λ Ɣ1 s)ɣ 1 s) ) 1/, λ R s Z\{} ˆψ k+ where Ɣ 1 s)sup λ R a Z ˆψ k+ λ a+1 ) ˆψ k+ λ A 1 inf ˆψ λ k+ λ R a Z a+1 B 1 sup ˆψ λ k+ λ R a Z a+1 s Z\{} + s Z\{} ˆψ k+ + πs a+1 b λ πs + a+1 b λ a+1 Ɣ1 s)ɣ 1 s) ) 1/ >, Ɣ1 s)ɣ 1 s) ) 1/ <. Similarly, we get the inequality a m ā l j i,k λ l, λ m ) a λ Ɣ s)ɣ s) ) 1/, λ R s Z\{} where Ɣ s)sup λ R a Z ˆψ i+ λ a+1 ) ˆψ k+ λ A inf λ ˆψ i+ λ R a Z a+1 B sup λ ˆψ i+ λ R a Z a+1 ˆψ k+ ˆψ k+ λ a+1 λ a+1 + πs a+1 b s Z\{} + s Z\{} ) 1/ ) 1/ ) and let us take ) and we take Ɣ s)ɣ s) ) 1/ >, Ɣ s)ɣ s) ) 1/ <. Using.7) and the constants A 1, B 1, A, B,wegettheinequality.6) for the trigonometric polynomials. Then we find the result for almost periodic functions by a standard approximation. Theorem 3 Let f be an almost periodic function and let α be an angle where cot α >. Then the BWT of f is a strong limit power function in y. Proof Let f t) n k1 a ke iλkt be a trigonometric polynomial. Then B ψ f )b, a) f t)ψ b,a t) dσ t) a k e iλkt γ +1)t γ ψ b,a t) γ +1/ Ɣγ + 3 ) dt γ +1) γ +1/ Ɣγ + 3 ) γ +1) γ +1/ Ɣγ + 3 ) γ +1) γ +1/ Ɣγ + 3 ) k1 a k t γ e iλkt ψ b,a t) dt k1 a k e iλ kv k1 a k e iλkv W φb,a v, λ k ), k1 v u v) γ e iλ ku ψ b,a u v) du

12 Ünalmış UzunJournal of Inequalities and Applications 15) 15:388 Page 1 of 1 where W φb,a isthewavepackettransform[8] with respect to the Bessel wavelet. Hence B ψ f )b, a) is a trigonometric polynomial in v. For a general almost periodic function f we take a sequence of trigonometric polynomials f n )suchthat f n f, and it will be sufficient to verify that B ψ f n B ψ f L v). Using the solutions in [], we have B ψ f n B ψ f L v) fn f ) φ L v) ess sup fn u v) f u v) ) ψ b,a u v) dσ u) v f n f ψ b,a 1, which gives the desired result. Competing interests The author declares that she has no competing interests. Received: 8 July 15 Accepted: 6 November 15 References 1. Arfken, GB, Weber, HJ, Harris, FE: Mathematical Methods for Physicists: A Comprehensive Guide. Academic Press, New York 1). Sneddon, IN: The Use of Integral Transforms. McGraw-Hill, New York 197) 3. Tuan, VK, Saigo, M: Convolution of Hankel transform and its application to an integral involving Bessel functions of first kind. Int. J. Math. Math. Sci. 183), ) 4. Kerr, FH: A fractional power theory for Hankel transforms. J. Math. Anal. Appl. 158, ) 5. Namias, V: Fractionalization of Hankel transform. J. Inst. Math. Appl. 6, ) 6. Prasad, A, Mahato, KL: The fractional Hankel wavelet transformation. Asian-Eur. J. Math. 8), ) 7. Sheppard, CJR, Larkin, KG: Similarity theorems for fractional Fourier transforms and fractional Hankel transforms. Opt. Commun. 154, ) 8. Taywade, RD, Gudadhe, AS, Mahalle, VN: Generalized operational relations and properties of fractional Hankel transform. Sci. Rev. Chem. Commun. 3), ) 9. Taywade, RD, Gudadhe, AS, Mahalle, VN: Initial and final value theorem on fractional Hankel transform. IOSR J. Math. 5, ) 1. Namias, V: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 5, ) 11. Ozaktas, HM, Zalvesky, Z, Kuntay, MA: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, Chichester 1) 1. Pei, S-C: Fractional cosine, sine, and Hartley transforms. IEEE Trans. Signal Process. 57), ) 13. Pathak, RS, Dixit, MM: Continuous and discrete Bessel wavelet transforms. J. Comput. Appl. Math. 161-), ) 14. Prasad, A, Mahato, A, Singh, VK, Dixit, MM: The continuous fractional Bessel wavelet transformation. Bound. Value Probl. 13,4 13) 15. Upadhyay, SK, Yadav, RN, Debnath, L: On continuous Bessel wavelet transformation associated with the Hankel-Hausdorff operator. Integral Transforms Spec. Funct. 35), ) 16. Poularikas, AD: Transforms and Applications Handbook. CRC Press, Boca Raton 1) 17. Gudadhe, AS, Taywade, RD, Mahalle, VN: Namias fractional Hankel transform in the Zemanian space. Glob. J. Math. Sci.: Theory Pract. 53), ) 18. Shen, L-C: Orthogonal polynomials on the unit circle associated with the Laguerre polynomials. Proc. Am. Math. Soc. 193), ) 19. Gradshteyn, IS, Ryzhik, IM: Tables of Integrals, Series and Products. Academic Press, New York ). Pathak, RS, Upadhyay, SK, Pandey, RS: The Bessel wavelet convolution product. Rend. Semin. Mat. Torino) 693), ) 1. Besicovitch, AS: Almost Periodic Functions. Dover, Cambridge 1954). Corduneanu, C: Almost Periodic Functions. Chelsea Publishing Company, New York 1989) 3. Zhang, C: Strong limit power functions. Studia Sci. Math. Hung. 13), ) 4. Zhang, C, Meng, C: Two new spaces of vector-valued limit power functions. Studia Sci. Math. Hung. 444), ) 5. Gudadhe, AS, Taywade, RD, Mahalle, VN: A relation between fractional Hankel transform and other integral transforms of fractional order. Asian J. Curr. Eng. Maths 5), ) 6. Jeffrey, A, Zwillinger, D: Table of Integrals, Series, and Products. Academic Press, New York 7) 7. Ünalmış Uzun, B: Generalized frames in the space of strong limit power functions. Mediterr. J. Math. 11), ) 8. Kato, K, Kobayashi, M, Ito, S: Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator. SUT J. Math. 47), )