Approximation of Discontinuous Signals by Exponential Sampling Series
AAPPROXIMATION OF DISCONTINUOUS SIGNALS BYEXPONENTIAL SAMPLING SERIESSATHISH KUMAR ANGAMUTHU , PRASHANT KUMAR andDEVARAJ PONNAIAN , Department of Mathematics, Visvesvaraya National Institute of Technology Nagpur,Nagpur, Maharashtra-440010, India [email protected] and [email protected] School of Mathematics, Indian Institute of Science Education and Research,Thiruvananthapuram, India. [email protected]
Abstract.
We analyse the behaviour of the exponential sampling series S χw f at jumpdiscontinuity of the bounded signal f. We obtain a representation lemma that is used foranalysing the series S χw f and we establish approximation of jump discontinuity functionsby the series S χw f. The rate of approximation of the exponential sampling series S χw f isobtained in terms of logarithmic modulus of continuity of functions and the round-offand time-jitter errors are also studied. Finally we give some graphical representation ofapproximation of discontinuous functions by S χw f using suitable kernels.Keywords: Exponential Sampling Series, Discontinuous Functions, Logarithmic Modulusof Smoothness, Rate of Approximation, Round-off and Time Jitter ErrorsMathematics Subject Classification(2010): 41A25, 26A15, 41A35. Introduction and Preliminaries
Let R + denote the set of all positive real numbers and let χ be a real valued functiondefined on R + . For ν ∈ N = N ∪ { } , the algebraic moments of order ν is defined by m ν ( χ, u ) := + ∞ (cid:88) k = −∞ χ ( e − k u )( k − log u ) ν , ∀ u ∈ R + . In a similar way, we can define the absolute moment of order ν as M ν ( χ, u ) := + ∞ (cid:88) k = −∞ | χ ( e − k u ) || k − log u | ν , ∀ u ∈ R + . We define M ν ( χ ) := sup u ∈ R + M ν ( χ, u ) . We say that χ is a kernel if it satisfies the followingconditions:(i) for every u ∈ R + , + ∞ (cid:88) k = −∞ χ ( e − k u ) = 1 , a r X i v : . [ m a t h . F A ] J a n (ii) for some ν > , M ν ( χ, u ) = sup u ∈ R + + ∞ (cid:88) k = −∞ | χ ( e − k u ) || k − log u | ν < + ∞ . Let Φ denote the set of all functions satisfying conditions (i) and (ii). For t ∈ R + , χ ∈ Φand w > , the exponential sampling series for a function f : R + → R is defined by ([6])( S χw f )( t ) = + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw ) . (1.1)It is easy to see that the series S χw f is well defined for f ∈ L ∞ ( R + ) . Using the abovesampling series S χw f one can reconstruct the functions which are not Mellin-band limited.Recently, Bardaro et.al. [4] pointed out that the study of Mellin-band limited functionsare different from that of Fourier-band limited functions. Mamedov was the first personwho studied the Mellin theory in [17] and then Butzer et.al. further developed the Mellin’stheory and studied its approximation properties in [10, 11, 9, 13]. The reconstruction us-ing exponential sampling formula was first studied by Butzer and Jansche in [9]. Thepointwise and uniform convergence of the series S χw f for continuous functions was anal-ysed in [6] and the convergence of S χw f was studied in Mellin-Lebesgue spaces. RecentlyBardaro et.al. studied various approximation results using Mellin transform which canbe seen in [5, 2, 3, 4, 6]. To improve the rate of convergence, a linear combination of S χw f was taken in [1]. The exponential sampling series with the sample points which areexponentially spaced on R + has been obtained as solution of some mathematical modelrelated to light scattering, Fraunhofer diffraction and radio astronomy (see [8, 15, 16, 18]).The approximation of discontinuous functions by classical sampling operators was firstinitiated by Butzer et.al.[12]. Further, the Kantorovich sampling series for discontinu-ous signals was analysed in [14]. Inspired by these works we analyse the behaviour ofexponential sampling series (1.1) as w → ∞ for discontinuous functions at the jumpdiscontinuities, i.e., at a point t where the limits f ( t + 0) := lim p → + f ( t + p ) , and f ( t −
0) := lim p → + f ( t − p )exists and are different. For a kernel χ, we define the functions ψ + χ ( u ) := (cid:88) k< log u χ ( ue − k ) , and ψ − χ ( u ) := (cid:88) k> log u χ ( ue − k ) . We observe that ψ + χ ( u ) and ψ − χ ( u ) are recurrent functions with fundamental interval[1 , e ]. Now we shall recall the definition of Mellin transform. Let L p ( R + ) , p ∈ [1 , ∞ ) bethe set of all Lebesgue measurable and p -integrable functions defined on R + . For c ∈ R ,we define the space X c = { f : R + → C : f ( · )( · ) c − ∈ L ( R + ) } equipped with the norm (cid:107) f (cid:107) X c = (cid:107) f ( · )( · ) c − (cid:107) = (cid:90) + ∞ | f ( y ) | y c − dy. For f ∈ X c , it’s Mellin transform is defined by (cid:100) [ f ] M ( s ) := (cid:90) + ∞ y s − f ( y ) dy , ( s = c + it, t ∈ R ) . We say that a function f ∈ X c ∩ C ( R + ) , c ∈ R is Mellin band-limited in the interval[ − κ, κ ] , if (cid:100) [ f ] M ( c + it ) = 0 for all | t | > κ, κ ∈ R + . The paper is organized as follows. In section 2, we prove the representation lemma forthe exponential sampling series (1.1) and using this lemma we analyse the approximationof discontinuous functions by S χw f in Theorem 2, Theorem 3 and Theorem 5. Further weanalyse the degree of approximation for the sampling series (1.1) in-terms of logarithmicmodulus of smoothness in section 3. In section 4, we study the round-off and time jittererrors for these sampling series. In example section, we have given a construction of afamily of Mellin-band limited kernels such that χ (1) = 0 for which S χw f converge at anyjump discontinuities. Further, the convergence at discontinuity points of the samplingseries S χw f has been tested numerically and numerical results are provided in Tables 1, 2and 3. 2. Approximation of Discontinuous Signals
For any given bounded function f : R + → R , we first prove the following representationlemma for the sampling series S χw f. Throughout this section we assume that the right andleft limits of f at t ∈ R + exist and are finite. Lemma 1.
For a given bounded function f : R + → R and a fixed t ∈ R + , let h t : R + → R be defined by h t ( x ) = f ( x ) − f ( t − , if x < tf ( x ) − f ( t + 0) , if x > t , if x = t. Then the following holds: ( S χw f )( t ) = ( S χw h t )( t ) + f ( t −
0) + ψ − χ ( t w )[ f ( t + 0) − f ( t − χ (1)[ f ( t ) − f ( t − , if w log( t ) ∈ Z and ( S χw f )( t ) = ( S χw h t )( t ) + f ( t −
0) + ψ − χ ( t w )[ f ( t + 0) − f ( t − , if w log( t ) / ∈ Z . Proof.
Let w log( t ) ∈ Z , and w > . Then, we can write( S χw h t )( t ) = (cid:88) k
0) + ψ − χ ( t w )[ f ( t + 0) − f ( t − χ (1)[ f ( t ) − f ( t − . Now let w log( t ) / ∈ Z and w > . Then repeating the same computations, we easily obtain( S χw f )( t ) = ( S χw h t )( t ) + f ( t −
0) + [ f ( t + 0) − f ( t − (cid:88) k ≥ w log( t ) χ ( e − k t w )= ( S χw h t )( t ) + f ( t −
0) + [ f ( t + 0) − f ( t − ψ − χ ( t w ) . (cid:3) Before proving the approximation of discontinuous functions by S χw f, we recall thefollowing theorem proved in ([6]) for continuous functions on R + . Theorem 1.
Let f : R + → R be a bounded function and χ ∈ Φ . Then ( S χw f )( t ) convergesto f ( t ) at any point t of continuity. Moreover, if f ∈ C ( R + ) , then we have lim w →∞ (cid:107) f − S χw f (cid:107) ∞ = 0 . Now we analyse the behaviour of the exponential sampling series at jump discontinuityat t ∈ R + when w log( t ) ∈ Z . Theorem 2.
Let f : R + → R be a bounded signal and let t ∈ R + be a point of non-removable jump discontinuity of f. For a given α ∈ R , the following statements are equiv-alent:(i) lim w →∞ w log( t ) ∈ Z ( S χw f )( t ) = αf ( t + 0) + [1 − α − χ (1)] f ( t −
0) + χ (1) f ( t ) , (ii) ψ − χ (1) = α, (iii) ψ + χ (1) = 1 − α − χ (1) . Proof.
First, we prove that ( i ) ⇐⇒ ( ii ). In view of the representation Lemma 1, we have( S χw f )( t ) = ( S χw h t )( t ) + f ( t −
0) + ψ − χ ( t w )[ f ( t + 0) − f ( t − χ (1)[ f ( t ) − f ( t − , for any w > w log( t ) ∈ Z . Since h t is bounded and continuous at zero andusing Theorem 1, we obtain lim w →∞ ( S χw h t )( t ) = 0 . Thus, we havelim w →∞ w log( t ) ∈ Z ( S χw f )( t ) = f ( t −
0) + (cid:32) lim w →∞ w log( t ) ∈ Z ψ − χ ( t w ) (cid:33) [ f ( t + 0) − f ( t − χ (1)[ f ( t ) − f ( t − . Now, we have lim w →∞ w log( t ) ∈ Z ψ − χ ( t w ) = lim w →∞ w log( t ) ∈ Z (cid:88) k>w log( t ) χ ( e − k t w ) . As ψ − χ is recurrent with fundamental domain [1 , e ] , we get ψ − χ ( t w ) = ψ − χ (1) , ∀ w, t such that w log( t ) ∈ Z . Therefore, we havelim w →∞ w log( t ) ∈ Z ( S χw f )( t ) = ψ − χ (1) f ( t + 0) + [1 − ψ − χ (1) − χ (1)] f ( t −
0) + χ (1) f ( t ) . Now ( i ) ⇐⇒ αf ( t + 0) + [1 − α − χ (1)] f ( t −
0) + χ (1) f ( t )= ψ − χ (1) f ( t + 0) + [1 − ψ − χ (1) − χ (1)] f ( t −
0) + χ (1) f ( t ) ⇐⇒ ψ − χ (1)( f ( t + 0) − f ( t − α ( f ( t + 0) − f ( t − ⇐⇒ ψ − χ (1) = α ⇐⇒ ( ii ) holds . Since + ∞ (cid:88) k = −∞ χ ( e − k t w ) = 1 , we have ψ + χ (1) = 1 − χ (1) − ψ − χ (1) . This implies that ( ii ) ⇐⇒ ( iii ) . Hence, the proof is completed. (cid:3)
Next we analyse the behaviour of the exponential sampling series at jump discontinuityat t ∈ R + when w log( t ) / ∈ Z . Theorem 3.
Let f : R + → R be a bounded signal and let t ∈ R + be a point of non-removable jump discontinuity of f. Let α ∈ R . Then the following statements are equiva-lent:(i) lim w →∞ w log( t ) / ∈ Z ( S χw f )( t ) = αf ( t + 0) + (1 − α ) f ( t − , (ii) ψ − χ ( u ) = α, u ∈ (1 , e ) (iii) ψ + χ ( u ) = 1 − α, u ∈ (1 , e ) . Proof.
Using representation Lemma 1, we obtainlim w →∞ w log( t ) / ∈ Z ( S χw f )( t ) = f ( t −
0) + (cid:32) lim w →∞ w log( t ) / ∈ Z ψ − χ ( t w ) (cid:33) [ f ( t + 0) − f ( t − i ) ⇐⇒ αf ( t + 0) + (1 − α ) f ( t −
0) = f ( t −
0) + (cid:32) lim w →∞ w log( t ) / ∈ Z ψ − χ ( t w ) (cid:33) [ f ( t + 0) − f ( t − ⇐⇒ α [ f ( t + 0) − f ( t − (cid:32) lim w →∞ w log( t ) / ∈ Z ψ − χ ( t w ) (cid:33) [ f ( t + 0) − f ( t − ⇐⇒ α = lim w →∞ w log( t ) / ∈ Z ψ − χ ( t w ) ⇐⇒ α = ψ − χ ( u ) , ∀ u ∈ (1 , e ) ⇐⇒ ( ii ) holds . Let w log( t ) / ∈ Z . Then, we have ψ + χ ( t w ) + ψ − χ ( t w ) = 1 . Thus, we obtain ( ii ) ⇐⇒ ψ + χ ( u ) = 1 − α, u ∈ (1 , e ) . (cid:3) The results in the above Theorem 3 was proved by assuming that ψ − χ ( u ) is constanton (1 , e ) . In what follows, we show that if ψ − χ ( u ) is not a constant on (1 , e ) , then theexponential sampling series can not converge at jump discontinuities. Theorem 4.
Let χ be a kernel such that ψ − χ ( u ) is not constant on (1 , e ) . Let f : R + → R be a bounded signal with a non-removable jump discontinuity at t ∈ R + . Then ( S χw f )( t ) does not converge point-wise at t .Proof. Suppose not. Then lim w →∞ w log( t ) / ∈ Z ( S χw f )( t ) = (cid:96), for some (cid:96) ∈ R + . By the uniqueness ofthe limit and Lemma 1, we obtain (cid:96) = f ( t −
0) + lim w →∞ ψ − χ ( t w )[ f ( t + 0) − f ( t − . Since f ( t + 0) − f ( t − (cid:54) = 0 , we obtain (cid:96) − f ( t − f ( t + 0) − f ( t −
0) = lim w →∞ ψ − χ ( t w ) . The above expression gives a contradiction. Indeed, iflim w →∞ ψ − χ ( t w ) = C, where C is a constant, then it fails to satisfy that ψ − χ is recurrent and not a constant,hence the theorem proved. (cid:3) Finally, the more general theorem of the exponential sampling series at jump disconti-nuity at t ∈ R + for any bounded signal can be proved. Theorem 5.
Let f : R + → R be a bounded signal and let t ∈ R + be a point of non-removable jump discontinuity of f. Let α ∈ R . Suppose that the kernel χ satisfies theadditional condition that χ (1) = 0 . Then, the following statements are equivalent:(i) lim w →∞ ( S χw f )( t ) = αf ( t + 0) + (1 − α ) f ( t − , (ii) ψ − χ ( u ) = α, u ∈ [1 , e ) (iii) ψ + χ ( u ) = 1 − α, u ∈ [1 , e ) . Moreover, if in addition we assume that χ is continuous on R + , then the above statementsare equivalent to the following statements:(iv) (cid:90) χ ( u ) u kπi duu = (cid:26) , if k (cid:54) = 0 α, if k = 0 (v) (cid:90) ∞ χ ( u ) u kπi duu = (cid:26) , if k (cid:54) = 01 − α, if k = 0 . Proof.
Proceeding along the lines proof of Theorem 2 and Theorem 3, we see that ( i ) , ( ii )and ( iii ) are equivalent. Let χ be continuous on R + and let χ ( u ) = (cid:26) χ ( u ) , for u < , for u ≥ . Then, we have ψ − χ ( u ) = (cid:88) k> log u χ ( ue − k ) = (cid:88) k ∈ Z χ ( ue − k ) . Therefore, ψ − χ is recurrent continuous function with the fundamental interval [1 , e ] . UsingMellin-Poisson summation formula, we obtain ψ − χ ( u ) = + ∞ (cid:88) k = −∞ (cid:92) [ χ ] M (2 kπi ) u − kπi = + ∞ (cid:88) k = −∞ (cid:18)(cid:90) χ ( u ) u kπi duu (cid:19) u − kπi . Therefore, we obtain ψ − χ ( u ) = α, ∀ u ∈ [1 , e ) ⇐⇒ (cid:100) [ χ ] M (2 kπi ) = (cid:26) , if k (cid:54) = 0 α, if k = 0 ⇐⇒ (cid:90) χ ( u ) u kπi duu = (cid:26) , if k (cid:54) = 0 α, if k = 0 . This implies that ( ii ) ⇐⇒ ( iv ) . Finally using the condition + ∞ (cid:88) k = −∞ χ ( e − k u ) = 1 ⇐⇒ (cid:100) [ χ ] M (2 kπi ) = (cid:26) , if k (cid:54) = 01 , if k = 0the equivalence between ( iv ) and ( v ) can be established easily. Thus the proof is com-pleted. (cid:3) Remark 1.
Let f : R + → R be a bounded signal with a removable discontinuity t ∈ R + ,i.e. f ( t + 0) = f ( t −
0) = (cid:96).
Then we have(i) lim w →∞ w log( t ) ∈ Z ( S χw f )( t ) = (cid:96) + χ (1)[ f ( t ) − (cid:96) ] , (ii) lim w →∞ w log( t ) / ∈ Z ( S χw f )( t ) = (cid:96), (iii) If χ (1) = 0 , then lim w →∞ ( S χw f )( t ) = (cid:96). Degree of Approximation
In this section, we estimate the order of convergence of the exponential sampling seriesby using the logarithmic modulus of continuity. Let C ( R + ) denote the space of all realvalued bounded continuous functions on R + equipped with the supremum norm (cid:107) f (cid:107) ∞ :=sup x ∈ R + | f ( x ) | . We say that a function f : R + → R is log-uniformly continuous if thefollowing hold: for a given (cid:15) > , there exists δ > | f ( p ) − f ( q ) | < (cid:15) whenever | log p − log q | < δ, for any p, q ∈ R + . The subspace consisting of all bounded log-uniformlycontinuous functions on R + is denoted by C ( R + ) . Let f ∈ C ( R + ) . Then the logarithmicmodulus of continuity is defined by ω ( f, δ ) := sup {| f ( p ) − f ( q ) | : whenever | log( p ) − log( q ) | ≤ δ, δ ∈ R + } . The logarithmic modulus of continuity satisfies the following properties:(a) ω ( f, δ ) → , as δ → . (b) ω ( f, cδ ) ≤ ( c + 1) ω ( f, δ ) , for every δ, c > . (c) | f ( p ) − f ( q ) | ≤ ω ( f, δ ) (cid:18) | log p − log q | δ (cid:19) . Further properties of logarithmic modulus of continuity can be seen in [2]. In the followingtheorem, we obtain the order of convergence for the exponential sampling series when M ν ( χ ) < ∞ for 0 < ν < . Theorem 6.
Let χ ∈ Φ be a kernel such that M ν ( χ ) < ∞ for < ν < and f ∈ C ( R + ) .Then for sufficiently large w > , the following hold: | ( S χw f )( t ) − f ( t ) | ≤ ω ( f, w − ν )[ M ν ( χ ) + 2 M ( χ )] + 2 ν +1 (cid:107) f (cid:107) ∞ M ν ( χ ) w − ν , for every t ∈ R + . Proof.
Let t ∈ R + be fixed. Then using the condition + ∞ (cid:88) k = −∞ χ ( e − k t w ) = 1 , we obtain | ( S χw f )( t ) − f ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12) + ∞ (cid:88) k = −∞ χ ( e − k t w )( f ( e kw ) − f ( t )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) < w + (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) ≥ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) | f ( e kw ) − f ( t ) | := I + I . Let 0 < ν < . Then we have ω (cid:18) f, (cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:19) ≤ ω (cid:18) f, (cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12) ν (cid:19) . Therefore, using the above inequality and the property ( c ) , we obtain I ≤ (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) < w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) ω (cid:18) f, (cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12) ν (cid:19) ≤ (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) < w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) (cid:18) w ν (cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12) ν (cid:19) ω ( f, w − ν ) ≤ (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) < w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) ω ( f, w − ν )+ (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) < w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) w ν (cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12) ν ω ( f, w − ν ) ≤ ω ( f, w − ν ) (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) < w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) + ω ( f, w − ν ) (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) < w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − w log t (cid:12)(cid:12)(cid:12) ν . In view of the conditions M ( χ ) and M ν ( χ ) , we easily obtain I ≤ ω ( f, w − ν )[ M ( χ ) + M ν ( χ )] . Now we estimate I . Since (cid:12)(cid:12) k − w log t (cid:12)(cid:12) ≥ w , we have1 (cid:12)(cid:12) k − w log t (cid:12)(cid:12) ν ≤ ν w − ν , < ν < . Hence, we obtain I ≤ (cid:107) f (cid:107) ∞ (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) ≥ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) ≤ (cid:107) f (cid:107) ∞ (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) ≥ w (cid:12)(cid:12) k − w log t (cid:12)(cid:12) ν (cid:12)(cid:12) k − w log t (cid:12)(cid:12) ν (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) ≤ ν +1 (cid:107) f (cid:107) ∞ w − ν (cid:88)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) ≥ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12)(cid:12)(cid:12) k − w log t (cid:12)(cid:12) ν ≤ ν +1 (cid:107) f (cid:107) ∞ w − ν M ν ( χ ) < ∞ . On combining the estimates I and I , we get the desired estimate. (cid:3) Round-Off and Time Jitter Errors
This section is devoted to analyse round off and time jitter errors connected withexponential sampling series (1.1). The round-off error arises when the exact sample values f ( e kw ) are replaced by approximate close ones ¯ f ( e kw ) in the sampling series (1.1). Let ξ k = f ( e kw ) − ¯ f ( e kw ) be uniformly bounded by ξ, i.e., | ξ k |≤ ξ, for some ξ > . We areinterested in analysing the error when f ( t ) is approximated by the following exponentialsampling series: ( S χw ¯ f )( t ) = + ∞ (cid:88) k = −∞ χ ( e − k t w ) ¯ f ( e kw ) . The total round-off or quantization error is defined by( Q ξ f )( t ) := | ( S χw f )( t ) − ( S χw ¯ f )( t ) | . Theorem 7.
For f ∈ C ( R + ) , the following hold:(i) (cid:107) ( Q ξ f ) (cid:107) C ( R + ) ≤ ξM ( χ ) (ii) (cid:107) f − S χw ¯ f (cid:107) C ( R + ) ≤ Cω (cid:18) f, w (cid:19) + ξM ( χ ) , where C = M ( χ ) + M ( χ ) . Proof.
The error in the approximation can be splitted as | f ( t ) − ( S χw ¯ f )( t ) | ≤ | f ( t ) − ( S χw f )( t ) | +( Q ξ f )( t ) := I + ( Q ξ f )( t ) . The term I is the error arising if the actual sample value is used and the total round-offor quantization error can be evaluated by (cid:107) ( Q ξ f ) (cid:107) C ( R + ) = sup t ∈ R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw ) − + ∞ (cid:88) k = −∞ χ ( e − k t w ) ¯ f ( e kw ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup t ∈ R + + ∞ (cid:88) k = −∞ (cid:12)(cid:12)(cid:12)(cid:12) ξ k χ ( e − k t w ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ξM ( χ ) . In view of Theorem 4 ([6], page no.7), we have I ≤ M ( χ ) ω ( f, δ ) + ω ( f, δ ) wδ M ( χ ) . On combining the estimates I and I , we get (cid:107) f − S χw ¯ f (cid:107) C ( R + ) ≤ (cid:18) M ( χ ) + M ( χ ) wδ (cid:19) ω ( f, δ ) + ξM ( χ ) . Choosing δ = 1 w , we obtain (cid:107) f − S χw ¯ f (cid:107) C ( R + ) ≤ Cω ( f, w ) + ξM ( χ ) , where C = M ( χ ) + M ( χ ) . Hence, the proof is completed. (cid:3)
The time-jitter error occurs when the function f ( t ) being approximated from sampleswhich are taken at perturbed nodes, i.e., the exact sample values f ( e kw ) are replaced by f ( e kw + (cid:37) k ) in the sampling series (1.1). So we are interested in analysing time jittererror and the approximation behaviour when f ( t ) is approximated by the sampling series + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw + (cid:37) k ) . We assume that the values (cid:37) k are bounded by a small number (cid:37), i.e., | (cid:37) k |≤ (cid:37), for all k ∈ Z and for some (cid:37) > . The total time jitter error is defined by J (cid:37) f ( t ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw ) − + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw + (cid:37) k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Theorem 8.
For f ∈ C (1) ( R + ) , the following hold:(i) (cid:107) J (cid:37) f (cid:107) C ( R + ) ≤ (cid:37) (cid:107) f (cid:48) (cid:107) C ( R + ) M ( χ ) (ii) (cid:13)(cid:13)(cid:13)(cid:13) f ( . ) − (cid:80) + ∞ k = −∞ χ ( e − k ( . ) w ) f ( e kw + (cid:37) k ) (cid:13)(cid:13)(cid:13)(cid:13) C ( R + ) ≤ Cω (cid:18) f, w (cid:19) + (cid:37) (cid:107) f (cid:48) (cid:107) C ( R + ) M ( χ ) , where C = M ( χ ) + M ( χ ) . Proof.
Applying the mean value theorem, error can be estimated by (cid:107) J (cid:37) f (cid:107) C ( R + ) ≤ sup k ∈ Z { sup t ∈ R + | f ( e kw ) − f ( e kw + (cid:37) k ) |} sup t ∈ R + + ∞ (cid:88) k = −∞ | χ ( e − k t w ) |≤ | (cid:37) k (cid:107) f (cid:48) (cid:107) C ( R + ) | sup t ∈ R + + ∞ (cid:88) k = −∞ | χ ( e − k t w ) | ≤ (cid:37) (cid:107) f (cid:48) (cid:107) C ( R + ) M ( χ ) . From the above estimates it is clear that the jitter error essentially depends on the smooth-ness of function f. For f ∈ C (1) ( R + ) , the associated approximation error is estimated by (cid:12)(cid:12)(cid:12)(cid:12) f ( t ) − + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw + (cid:37) k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) f ( t ) − + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw ) − + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw + (cid:37) k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | f ( t ) − S χw f ( t ) | + J (cid:37) f ( t ) . Again using Theorem 4 ([6], page no.7), we have | f ( t ) − S χw f ( t ) | ≤ M ( χ ) ω ( f, δ ) + ω ( f, δ ) wδ M ( χ ) . Using the above estimate and J (cid:37) f, we obtain (cid:13)(cid:13)(cid:13)(cid:13) f ( . ) − + ∞ (cid:88) k = −∞ χ ( e − k ( . ) w ) f ( e kw + (cid:37) k ) (cid:13)(cid:13)(cid:13)(cid:13) C ( R + ) ≤ M ( χ ) ω ( f, δ ) + ω ( f, δ ) wδ M ( χ ) + (cid:37) (cid:107) f (cid:48) (cid:107) C ( R + ) M ( χ ) . Hence, the proof is completed. (cid:3) Examples of the Kernels
In this section, we provide certain examples of the kernel functions which will satisfyour assumptions. First we give the family of Mellin-B spline kernels. The Mellin B-splineof order n is given by¯ B n ( x ) := 1( n − n (cid:88) j =0 ( − j (cid:18) nj (cid:19)(cid:18) n x − j (cid:19) n +1+ , x ∈ R + It can be easily seen that ¯ B n ( x ) is compactly supported for every n ∈ N . The Mellintransform of ¯ B n (see [6]) is (cid:92) [ ¯ B n ] M ( c + it ) = (cid:18) sin( t )( t ) (cid:33) n , t (cid:54) = 0 . The Mellin’s-Poisson summation formula [11]) is defined by( i ) j + ∞ (cid:88) k = −∞ χ ( e k x )( k − log u ) j = + ∞ (cid:88) k = −∞ d j dt j (cid:100) [ χ ] M (2 kπi ) x − kπi , for k ∈ Z . We need the following lemma (see [6]).
Lemma 2.
The condition + ∞ (cid:88) k = −∞ χ ( e − k x w ) = 1 is equivalent to (cid:100) [ χ ] M (2 kπi ) = (cid:26) , if k = 00 , otherwise.Moreover m j ( χ, u ) = 0 for j = 1 , , · · · , n is equivalent to d j dt j (cid:100) [ χ ] M (2 kπi ) = 0 for j =1 , , · · · , n and ∀ k ∈ Z . Using the above Lemma, we obtain (cid:92) [ ¯ B n ] M (2 kπi ) = (cid:26) , if k = 00 , otherwise.Using Mellin’s-Poisson summation formula, it is easy to see that ¯ B n ( x ) satisfies the con-dition (i). As ¯ B n ( x ) is compactly supported, the condition (ii) is also satisfied. Next we consider the Mellin Jackson kernels. For x ∈ R + , β ∈ N , γ ≥ , the Mellin Jackson kernelsare defined by J − γ,β ( x ) := d γ,β x − c sinc β (cid:18) log x γβπ (cid:19) , where d − γ,β := (cid:90) ∞ sinc β (cid:18) log x γβπ (cid:19) duu . One can easily verify that the Mellin Jackson kernels also satisfies conditions (i) and (ii)(see [6]). We can analyse the convergence of the exponential sampling series with jumpdiscontinuity associated with these kernels only for the case given in Theorem 2 and weobserve that χ (1) (cid:54) = 0. So Theorem 5 can not be applied for these kernels. In order toobtain the convergence of the exponential sampling series at jump discontinuity t ∈ R + of the given bounded signal f : R + → R , we need to construct suitable kernels. One suchconstruction is given in the following theorem. Theorem 9.
Let χ a , χ b be two continuous kernels supported respectively in the intervals [ e − a , e a ] and [ e − b , e b ] . Let α ∈ R be fixed. We define χ : R + → R + by χ ( u ) := (1 − α ) χ a (2 ue − a − ) + αχ b (2 ue b ) , u ∈ R + . Then χ is a kernel satisfying conditions (i), (ii) and χ (1) = 0 . Moreover, the correspond-ing exponential sampling series S χw f, w > based upon χ satisfy (i) of Theorem 5 withparameter α for a given bounded signal f : R + → R at any non-removable discontinuity t ∈ R + of f. Proof.
The Mellin transform of χ ( u ) is (cid:100) [ χ ] M ( s ) = (cid:90) ∞ (1 − α ) χ a (2 te − a − ) t s − dt + (cid:90) ∞ αχ b (2 te b ) t s − dt = (1 − α ) (cid:92) [ χ a ] M ( s ) (cid:18) e (1+ a ) (cid:19) s + α (cid:91) [ χ b ] M ( s ) (cid:18) e − b (cid:19) s . It is simple to check that χ satisfies condition (ii). Now we show that kernel satisfies thecondition (i). We obtain (cid:100) [ χ ] M (2 kπi ) = (1 − α ) (cid:92) [ χ a ] M (2 kπi ) (cid:18) e (1+ a ) (cid:19) kπi + α (cid:91) [ χ b ] M (2 kπi ) (cid:18) e − b (cid:19) kπi . As χ a and χ b satisfies condition (i), we have (cid:92) [ χ a ] M (2 kπi ) = (cid:91) [ χ b ] M (2 kπi ) = (cid:26) , if k (cid:54) = 01 , if k = 0For suitable choices of a and b , we obtain (cid:100) [ χ ] M (2 kπi ) = (cid:26) , if k (cid:54) = 01 , if k = 0 . Therefore, χ satisfies condition (i) and we can easily see that χ (1) = 0 . Now, we obtain (cid:90) χ ( u ) u kπi − du = α (cid:90) χ b (2 e b u ) u kπi − du = α (cid:91) [ χ b ] M (2 kπi ) (cid:18) e − b kπi kπi (cid:19) = (cid:26) , if k (cid:54) = 0 α, if k = 0 . Therefore, the condition (iv) of Theorem 5 is satisfied, hence the proof is completed. (cid:3)
Now we test numerically the approximation of discontinuous function f ( t ) = t + 1 , t < , ≤ t < , ≤ t < t , t ≥ t = 32 , t = 72 and t = 112 . Weconsider a linear combination of Mellin B-spline kernels defined by (see Fig. 1) χ ( t ) = 14 ¯ B (2 te − ) + 34 ¯ B (2 te ) , where ¯ B is given by ¯ B ( t ) = − log t, < t < e t, e < t < , otherwise.Clearly the exponential sampling series S χw f based on χ ( t ) satisfies the conditions (i),(ii) and χ (1) = 0 . We also observe that the condition (i) of Theorem 5 is satisfied with α = 34 . From Theorem 5 and Theorem 9, we have that at the discontinuity points of f, thesampling series S χw f converges to 34 f ( t + 0) + 14 f ( t − . The convergence of the samplingseries S χw f at discontinuity points t = 32 , t = 72 and t = 112 of the function f has beentested and numerical results are presented in Tables 1, 2 and 3. Figure 1.
Plot of the kernel χ ( t ) = 14 ¯ B (2 te − ) + 34 ¯ B (2 te ) . Table 1.
Approximation of f at the jump discontinuity point t = 32 by the exponentialsampling series S χw f based on χ ( t ) for different values of w > . The theoretical limit of ( S χw f ) (cid:18) (cid:19) as w → ∞ is f (cid:18)
32 + 0 (cid:19) + 14 f (cid:18) − (cid:19) = 2 . .w S χw f . . . . . . Table 2.
Approximation of f at the jump discontinuity point t = 72 by the exponentialsampling series S χw f based on χ ( t ) for different values of w > . The theoretical limit of ( S χw f ) (cid:18) (cid:19) as w → ∞ is f (cid:18)
72 + 0 (cid:19) + 14 f (cid:18) − (cid:19) = 2 . .w S χw f .
25 2 .
25 2 .
25 2 .
25 2 .
25 2 . Table 3.
Approximation of f at the jump discontinuity point t = 112 by the exponentialsampling series S χw f based on χ ( t ) for different values of w > . The theoretical limit of ( S χw f ) (cid:18) (cid:19) as w → ∞ is f (cid:18)
112 + 0 (cid:19) + 14 f (cid:18) − (cid:19) = 1 . . Figure 2.
Approximation of f ( t ) by S χw f based on χ ( t ) = 14 ¯ B (2 te − ) +34 ¯ B (2 te ) for w = 5 .w S χw f . . . . . . Figure 3.
Approximation of f ( t ) by S χw f based on χ ( t ) = 14 ¯ B (2 te − ) +34 ¯ B (2 te ) for w = 10 . Acknowledgments.
The first two authors are supported by DST-SERB, India Re-search Grant EEQ/2017/000201. The third author P. Devaraj has been supported byDST-SERB Research Grant MTR/2018/000559.