Approximation of discontinuous functions by Kantorovich exponential sampling series
AAPPROXIMATION OF DISCONTINUOUS FUNCTIONS BYKANTOROVICH EXPONENTIAL SAMPLING SERIESA. Sathish Kumar , Prashant Kumar and P. Devaraj , 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.
The Kantorovich exponential sampling series I χw f at jump discontinuities ofthe bounded measurable signal f : R + → R has been analysed. A representation lemmafor the series I χw f is established and using this lemma certain approximation theoremsfor discontinuous signals are proved. The degree of approximation in terms of logarith-mic modulus of smoothness for the series I χw f is studied. Further a linear prediction ofsignals based on past sample values has been obtained. Some numerical simulations areperformed to validate the approximation of discontinuous signals f by I χw f. Keywords: Kantorovich Exponential Sampling Series, Discontinuous Signals, Logarith-mic Modulus of Smoothness, Mellin Transform, Degree of Approximation.Mathematics Subject Classification(2010): 41A35, 41A25, 26A15. Introduction
Preliminaries and Basic Assumptions.
First we give some basic definitions andnotations. Let L ∞ ( R + ) denote the set of all Lebesgue measurable essentially boundedfunctions on the set of all positive real numbers R + . For c ∈ R , we define X c = { f : R + → C : f ( · )( · ) c − ∈ L ( R + ) } and the norm on X c is given by (cid:107) f (cid:107) X c = (cid:107) f ( · )( · ) c − (cid:107) = (cid:90) + ∞ | f ( v ) | v c − dv. The Mellin transform of a function f ∈ X c is defined as (cid:100) [ f ] M ( s ) := (cid:90) + ∞ v s − f ( v ) dv , ( s = c + it, t ∈ R )and that it is said to be Mellin band-limited on [ − κ, κ ] , if (cid:100) [ f ] M ( c + it ) = 0 for all | t | > κ, where κ is some positive real number. Butzer and Jansche in [14] have initiated the studyof exponential sampling. For a suitable kernel χ : R + → R satisfying certain conditions, a r X i v : . [ m a t h . F A ] F e b Bardaro et.al. [6] have analysed the uniform and point-wise convergence of the followinggeneralized sampling series for bounded continuous functions:( S χw f )( t ) = + ∞ (cid:88) k = −∞ χ ( e − k t w ) f ( e kw ) (1.1)The above sampling series S χw f is well defined for f ∈ L ∞ ( R + ) . In [8], the exponentialsampling series (1.1) was studied for functions in Mellin-Lebesgue spaces. The Mellintheory received a lot of attentions due to analysis by Mamedov [26]. Further it wasdeveloped by Butzer et.al. in various research articles (see [15] -[20]). Some of the workwhich are related to Mellin transform and Mellin approximation can be seen in [5, 2, 3, 4].Bardaro et.al. [4] analysed the analogue of Paley-Wiener theorem for Mellin transformand pointed out that it is different from that of Fourier transform. The exponentialsampling series is used in certain areas like light scattering, Fraunhofer diffraction andradio astronomy, etc (see [12, 22, 24, 27]).Let χ : R + → R . For ν ∈ N := N ∪ { } , the algebraic and absolute moment of order ν are defined by m ν ( χ, u ) := + ∞ (cid:88) k = −∞ χ ( e − k u )( k − log u ) ν , ∀ u ∈ R + and 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 kernel if it satisfies the followingconditions:(i) χ ∈ L ( R + ) and χ is bounded on (cid:20) e , e (cid:21) , (ii) for every u ∈ R + , + ∞ (cid:88) k = −∞ χ ( e − k u ) = 1 , (iii) for some ν > , M ν ( χ, u ) = sup u ∈ R + + ∞ (cid:88) k = −∞ | χ ( e − k u ) || k − log u | ν < + ∞ . Kantorovich Exponential Sampling Series.
The exponential sampling series S χw f is used to approximate a non-Mellin band limited function f using its values atthe nodes ( e kw ) . Measuring the exact value of the sample is difficult and measurementprocess depends on the aperture device used for sampling. A suitable mathematicalmodel for such a measurement process is replacing f ( e kw ) by the local average of signal f on the interval (cid:104) e kw , e k +1 w (cid:105) . This motivates the consideration of Kantorovich version of theexponential sampling series. In 2020, Kantorovich exponential sampling series S χw f wasintroduced and their inverse and direct approximation results have been studied in [29]. For f : R + → R , the Kantorovich exponential sampling series of f is defined by (see [29])( I χw f )( x ) = + ∞ (cid:88) k = −∞ χ ( e − k x w ) w (cid:90) k +1 wkw f ( e u ) du. (1.2)Under the conditions (i) (ii) and (iii) on the kernel χ, it is clear that I χw f is well definedfor f ∈ L ∞ ( R + ) at every x ∈ R + . Direct convergence results and the improved orderof approximation for Kantorovich exponential sampling series has been analysed in [31].Further, the Durrmeyer type modification of exponential sampling series (1.1) is consid-ered in [1] and [30]. So far the approximation of continuous functions by Kantorovichexponential sampling series I χw f has been studied. The Kantorovich exponential samplingseries I χw f for discontinuous signals has not been analysed. Motivated by Butzer et.al.[21]and [23], we study the behaviour of Kantorovich exponential sampling series I χw f at jumpdiscontinuity of bounded measurable signal f. Approximation of Discontinuous Functions by I χw f We analyse the behaviour of the Kantorovich exponential sampling series I χw f as w → ∞ at a jump discontinuity of f , i.e., at a point where the limits f ( t + 0) := lim y → + f ( t + y ) , and f ( t −
0) := lim y → + f ( t − y )exist and are different. We denote ψ + χ and ψ − χ two functions from R + to R which aredefined by ψ + χ ( u ) := (cid:88) k< log u χ ( ue − k ) , and ψ − χ ( u ) := (cid:88) k> log u χ ( ue − k ) , where χ is a kernel function and one can easily verify that ψ + χ ( u ) and ψ − χ ( u ) are recurrentfunctions with fundamental interval [1 , e ]. We first prove the following representationlemma for the series I χw f. Throughout this section we assume that f ( t + 0) and f ( t − Lemma 1.
Let f : R + → R be a bounded measurable signal and t ∈ R + be fixed. 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, there holds: ( I χw f )( t ) = ( I χw h t )( t ) + f ( t −
0) + (cid:2) χ ( e w log t −(cid:98) w log t (cid:99) ) + ψ − χ ( t w ) (cid:3) ( f ( t + 0) − f ( t − − χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) ( w log t − (cid:98) w log t (cid:99) ) ( f ( t + 0) − f ( t − , if w log( t ) / ∈ Z , where (cid:98) . (cid:99) denotes the integer part of a given number and ( I χw f )( t ) = ( I χw h t )( t ) + f ( t −
0) + (cid:0) χ (1) + ψ − χ ( t w ) (cid:1) [ f ( t + 0) − f ( t − , if w log( t ) ∈ Z , w > . Proof.
Let w log( t ) / ∈ Z , and w > . Then, we have( I χw h t )( t ) = (cid:88) k< (cid:98) w log t (cid:99) χ ( e − k t w ) w (cid:90) k +1 wkw ( f ( e u ) − f ( t − du + (cid:88) k> (cid:98) w log t (cid:99) χ ( e − k t w ) w (cid:90) k +1 wkw ( f ( e u ) − f ( t + 0)) du + χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) w (cid:32)(cid:90) log t (cid:98) w log t (cid:99) w ( f ( e u ) − f ( t − du + (cid:90) (cid:98) w log t (cid:99) +1 w log t ( f ( e u ) − f ( t + 0)) du (cid:33) = (cid:88) k< (cid:98) w log t (cid:99) χ ( e − k t w ) w (cid:90) k +1 wkw f ( e u ) du − f ( t − (cid:88) k< (cid:98) w log t (cid:99) χ ( e − k t w )+ (cid:88) k> (cid:98) w log t (cid:99) χ ( e − k t w ) w (cid:90) k +1 wkw f ( e u ) du − f ( t + 0) (cid:88) k> (cid:98) w log t (cid:99) χ ( e − k t w )+ χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) w (cid:90) (cid:98) w log t (cid:99) +1 w (cid:98) w log t (cid:99) w f ( e u ) du − χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) × w (cid:18) f ( t − (cid:18) log t − (cid:98) w log t (cid:99) w (cid:19) + f ( t + 0) (cid:18) (cid:98) w log t (cid:99) + 1 w − log t (cid:19)(cid:19) = ( I χw f )( t ) − f ( t − (cid:88) k< (cid:98) w log t (cid:99) χ ( e − k t w ) − f ( t + 0) (cid:88) k> (cid:98) w log t (cid:99) χ ( e − k t w ) − χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) [ f ( t − w log t − (cid:98) w log t (cid:99) ) + f ( t + 0)( (cid:98) w log t (cid:99) + 1 − w log t )] . Adding and subtracting f ( t − (cid:88) k ≥(cid:98) w log( t ) (cid:99) χ ( e − k t w ) in the above equation, we obtain( I χw f )( t ) = ( I χw h t )( t ) + f ( t − (cid:88) k ∈ Z χ ( e − k t w ) − f ( t − (cid:88) k> (cid:98) w log t (cid:99) χ ( e − k t w ) − f ( t − χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) + f ( t + 0) χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) + f ( t + 0) (cid:88) k> (cid:98) w log t (cid:99) χ ( e − k t w )+ χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) (cid:104) f ( t − w log t − (cid:98) w log t (cid:99) ) + f ( t + 0)( (cid:98) w log t (cid:99) − w log t ) (cid:105) . As w log t is not an integer, one can easily verify that ψ − χ ( t w ) = (cid:88) k> (cid:98) w log t (cid:99) χ ( e − k t w )and in view of condition (ii) we obtain( I χw f )( t ) = ( I χw h t )( t ) + f ( t −
0) + (cid:2) χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) + ψ − χ ( t w ) (cid:3) ( f ( t + 0) − f ( t − − χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) ( w log t − (cid:98) w log t (cid:99) ) ( f ( t + 0) − f ( t − . Now we analyse the case that w log( t ) ∈ Z , and w > . We have( I χw h t )( t ) = (cid:88) k 0) + ( f ( t + 0) − f ( t − (cid:88) k ≥ w log( t ) χ ( e − k t w )= ( I χw h t )( t ) + f ( t − 0) + (cid:0) χ (1) + ψ − χ ( t w ) (cid:1) [ f ( t + 0) − f ( t − . (cid:3) We recall the following basic convergence result for continuous functions proved in [29]. Theorem 1. Let f : R + → R be a bounded function . Then, ( I χw f )( t ) converges to f ( t ) at any point t of continuity. Further, if f ∈ C ( R + ) , then lim w →∞ (cid:107) f − I χw f (cid:107) ∞ = 0 . Now we show that the non-removable jump discontinuity function can be approximatedby Kantorovich exponential sampling series I χw f for w log( t ) ∈ Z . Theorem 2. Suppose that f has a non-removable jump discontinuity at t ∈ R + and that α ∈ R . Then, the following are equivalent:(i) lim w →∞ w log( t ) ∈ Z ( I χw f )( t ) = [ χ (1) + α ] f ( t + 0) + [1 − α − χ (1)] f ( t − , (ii) ψ − χ (1) := α, (iii) ψ + χ (1) := 1 − α − χ (1) . Proof. Let w log( t ) ∈ Z . In view of Lemma 1, we have( I χw f )( t ) = ( I χw h t )( t ) + f ( t − 0) + ( χ (1) + ψ − χ ( t w ))[ f ( t + 0) − f ( t − , for any w > . Since h t is bounded and continuous at zero and applying Theorem 1, weget lim w →∞ ( I χw h t )( t ) = 0 . Hence, we havelim w →∞ w log( t ) ∈ Z ( I χw f )( t ) = f ( t − 0) + (cid:32) χ (1) + lim w →∞ w log( t ) ∈ Z ψ − χ ( t w ) (cid:33) [ f ( t + 0) − f ( t − . Since ψ − χ is recurrent function with fundamental domain [1 , e ] , we get ψ − χ ( t w ) = ψ − χ (1) , ∀ w, t such that w log( t ) ∈ Z . Thus, we obtainlim w →∞ w log( t ) ∈ Z ( I χw f )( t ) = [ χ (1) + ψ − χ (1)] f ( t + 0) + [1 − ψ − χ (1) − χ (1)] f ( t − . Now ( i ) ⇐⇒ [ χ (1) + α ] f ( t + 0) + [1 − α − χ (1)] f ( t − χ (1) + ψ − χ (1)] f ( t + 0) + [1 − ψ − χ (1) − 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 in order to obtain the convergence of Kantorovich exponential sampling series I χw f at a non-removable jump discontinuity at t ∈ R + when w log( t ) / ∈ Z an extra conditionon the kernel is required. We prove the following theorem. Theorem 3. Let f : R + → R be a bounded measurable signal with a non-removable jumpdiscontinuity at t ∈ R + and let α ∈ R . Suppose that χ ( u ) = 0 , for every u ∈ (1 , e ) . Then,the following statements are equivalent:(i) lim w →∞ w log( t ) / ∈ Z ( I 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 Lemma 1 and the condition that χ ( u ) = 0 , for every u ∈ (1 , e ) , we obtainlim w →∞ w log( t ) / ∈ Z ( I 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 . Since ψ + χ and ψ − χ are recurrent functions on [1 , e ] , we have ψ + χ ( t w ) + ψ − χ ( t w ) = 1 . Hence, we obtain ( ii ) ⇐⇒ ψ + χ ( u ) = 1 − α, u ∈ (1 , e ) . (cid:3) The above Theorem is proved by assuming that χ ( u ) = 0 , for every u ∈ (1 , e ) . In thefollowing theorem we prove that without this condition, it is not possible to show theconvergence of I χw f at jump discontinuities. Theorem 4. Let χ be a kernel such that ψ − χ ( u ) = α for every u ∈ (1 , e ) and χ ( u ) (cid:54) = 0 , for some u ∈ (1 , e ) . Let f : R + → R be a bounded measurable signal with a non-removablejump discontinuity t ∈ R + . Then ( I χw f )( t ) does not converge point-wise at t .Proof. Suppose on contrary that ( I χw f )( t ) converge point-wise at t, that is lim w →∞ w log( t ) / ∈ Z ( I χw f )( t ) = (cid:96), for some (cid:96) ∈ R + . Using the existence of the limit and Lemma 1, we obtain (cid:96) = f ( t − 0) + (cid:32) lim w →∞ w log( t ) / ∈ Z (cid:2) χ ( e w log t −(cid:98) w log t (cid:99) ) + ψ − χ ( t w ) (cid:3)(cid:33) ( f ( t + 0) − f ( t − − [ f ( t + 0) − f ( t − (cid:32) lim w →∞ w log( t ) / ∈ Z χ (cid:0) e w log t −(cid:98) w log t (cid:99) (cid:1) ( w log t − (cid:98) w log t (cid:99) ) (cid:33) . Since ψ − χ ( u ) = α for every u ∈ (1 , e ) , ∀ w > , w log( t ) / ∈ Z , we have w log t − (cid:98) w log t (cid:99) = ξ ∈ (1 , e ) , so we obtain (cid:96) = f ( t − 0) + [ f ( t + 0) − f ( t − α + χ ( ξ )) − [ f ( t + 0) − f ( t − ξχ ( ξ ) , ∀ ξ ∈ (1 , e ) . Since f has a jump discontinuity t ∈ R + , we obtain χ ( ξ ) = (cid:18) (cid:96) − f ( t − f ( t + 0) − f ( t − − α (cid:19) (cid:18) − ξ (cid:19) , ∀ ξ ∈ (1 , e ) . The above expression gives a contradiction. Indeed, if (cid:96) − f ( t − f ( t + 0) − f ( t − − α := C (cid:54) = 0 , then χ ( ξ ) is unbounded on (1 , e ) , so it fails to satisfy condition (i). If C = 0 , then χ ( ξ ) = 0 , for every ξ ∈ (1 , e ) , this is again a contradiction. Hence the proof is completed. (cid:3) Next we prove a more general theorem on the convergence of the sampling series I χw f for any bounded signal f at jump discontinuities. Theorem 5. Let f : R + → R be a bounded measurable signal and let t ∈ R + be anon-removable jump discontinuity point of f. For any α ∈ R and any kernel function χ satisfying the additional condition χ ( u ) = 0 , for every u ∈ [1 , e ) , the following areequivalent:(i) lim w →∞ ( I χw f )( t ) = αf ( t + 0) + (1 − α ) f ( t − , (ii) ψ − χ ( u ) := α, for every u ∈ [1 , e ) (iii) ψ + χ ( u ) := 1 − α, for every u ∈ [1 , e ) . If we further assume that χ is continuous on R + , then the above statements are alsoequivalent to the following:(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. Now we assume that χ is continuous on R + . Define 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 )is a recurrent continuous function with the fundamental interval [1 , e ] . By the MellinPoisson’s summation formula, we get 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 . Hence, we have ( ii ) ⇐⇒ ( iv ) . 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. (cid:3) Remark 1. If f : R + → R be a bounded measurable signal with a removable discontinuity t ∈ R + , then we have lim w →∞ w log( t ) ∈ Z ( I χw f )( t ) = lim w →∞ w log( t ) / ∈ Z ( I χw f )( t ) = lim w →∞ ( I χw f )( t ) = (cid:96). Degree of Approximation We study the degree of approximation of I χw f by using the logarithmic modulus ofcontinuity. A function f : R + → R is said to be log-uniformly continuous if for every (cid:15) > , ∃ δ > | f ( s ) − f ( t ) | < (cid:15) whenever | log s − log t | < δ, for any s, t ∈ R + . The set of all bounded log-uniformly continuous functions is C ( R + ) . It is easy to see that C ( R + ) is contained in C ( R + ) , where C ( R + ) is the set of all bounded continuous functionson R + with usual sup norm (cid:107) f (cid:107) ∞ := sup x ∈ R + | f ( x ) | . For f ∈ C ( R + ) , the logarithmicmodulus of continuity is defined by ω ( f, δ ) := sup {| f ( s ) − f ( t ) | : whenever | log( s ) − log( t ) | ≤ δ, δ ∈ R + } .ω ( f, δ ) has the following properties:(a) ω ( f, δ ) → , as δ → . (b) | f ( s ) − f ( t ) | ≤ ω ( f, δ ) (cid:18) | log s − log t | δ (cid:19) . Now we show the order of convergence for the Kantorovich exponential sampling serieswhen M ν ( χ ) < + ∞ for 0 < ν < . Theorem 6. Let χ be a kernel such that M ν ( χ ) < + ∞ for < ν < . Then for any f ∈ C ( R + ) and for sufficiently large w > , we have | ( I 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 | ( I χw f )( t ) − f ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ (cid:88) k = −∞ χ ( e − k t w ) w (cid:90) k +1 wkw ( f ( e u ) − f ( t )) du (cid:12)(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) w (cid:90) k +1 wkw | f ( e u ) − f ( t ) | du := I + I . Let u ∈ (cid:2) kw , k +1 w (cid:3) . If | w log t − k | ≤ w , we have | u − log t | ≤ (cid:12)(cid:12) u − kw (cid:12)(cid:12) + (cid:12)(cid:12) kw − log t (cid:12)(cid:12) ≤ w + ≤ . Let 0 < ν < w > ω ( f, | u − log t | ) ≤ ω ( f, | u − log t | ν ) . Therefore, using the above inequality and the property ( b ) , we obtain I ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) w (cid:90) k +1 wkw ω ( f, | u − log t | ν ) du ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) w (cid:90) k +1 wkw (1 + w ν | u − log t | ν ) ω ( f, w − ν ) du ≤ ω ( f, w − ν ) (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) w (cid:90) k +1 wkw w ν | u − w log t | ν du + (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) =: I (cid:48) + I (cid:48)(cid:48) . Using the condition ( ii ) we easily obtain I (cid:48)(cid:48) ≤ m ( χ ) . Utilizing the sub-additivity of | . | ν for 0 < ν < 1, we have: I (cid:48) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) (cid:32) w ν max u ∈ [ kw , k +1 w ] | u − log t | ν (cid:33) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) w ν (cid:18) max (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν , (cid:12)(cid:12)(cid:12)(cid:12) k + 1 w − log t (cid:12)(cid:12)(cid:12)(cid:12) ν (cid:19)(cid:19) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) (cid:18) w ν max (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν , (cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν + 1 w ν (cid:19)(cid:19) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) (cid:18) w ν (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν + 1 w ν (cid:19)(cid:19) ≤ M ν ( χ ) + M ( χ ) < + ∞ . Finally we estimate I : 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 result. (cid:3)1 1, we have: I (cid:48) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) (cid:32) w ν max u ∈ [ kw , k +1 w ] | u − log t | ν (cid:33) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) w ν (cid:18) max (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν , (cid:12)(cid:12)(cid:12)(cid:12) k + 1 w − log t (cid:12)(cid:12)(cid:12)(cid:12) ν (cid:19)(cid:19) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) (cid:18) w ν max (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν , (cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν + 1 w ν (cid:19)(cid:19) ≤ (cid:88) | k − w log t |≤ w (cid:12)(cid:12) χ ( e − k t w ) (cid:12)(cid:12) (cid:18) w ν (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) kw − log t (cid:12)(cid:12)(cid:12)(cid:12) ν + 1 w ν (cid:19)(cid:19) ≤ M ν ( χ ) + M ( χ ) < + ∞ . Finally we estimate I : 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 result. (cid:3)1 Linear Prediction of Signals The Kantorovich exponential sampling series can be used for predicting the signals ata time t using the sample values taken in the past. This is addressed in the followingtheorem. Theorem 7. Let χ be a kernel with compact support and f : R + → R be a boundedmeasurable signals and w > . If supp χ ⊂ (1 , ∞ ) , then for every w > and t ∈ R + with w log t ∈ Z , we have ( I χw f )( t ) = (cid:88) k Let w > . As supp χ ⊂ (1 , ∞ ) , we obtain χ ( e − k t w ) = 0 , for every k ∈ Z such that e − k t w ≤ , this implies that log t ≤ kw . If w log t ∈ Z , it is easy to observe that the lastterm of the Kantorovich exponential sampling series is χ ( e ) (cid:32) w (cid:90) log t log t − w f ( e u ) du (cid:33) . Therefore, we see that Kantorovich exponential sampling series exploit the values of thesignal f in the past with respect to fixed time t. Now the second case can be estimated as follows: Let supp χ ⊂ ( e, ∞ ) . If w log t / ∈ Z then (cid:98) w log t (cid:99) < w log t < (cid:98) w log t (cid:99) + 1. Finally, we have (cid:88) k Remark 2. It is observed that in order to approximate the signal at a time t, the localaverage values of the signals are computed before the time instance t. The above theoremcan be used for predicting the future signal values from the past value of the signal. Examples of the Kernels and Special Cases Mellin-B spline kernels. The Mellin B-splines of order n for x ∈ R + are 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+ . ¯ B n ( x ) is compactly supported for every n ∈ N . So it is bounded and it belongs to L ( R + ) . Therefore condition (i) is satisfied. The Mellin transform 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 . Using the Mellin’s-Poisson summation formula (see [6]), it is easy to see that ¯ B n ( x )satisfies the condition (ii). The condition (iii) is also easily verified.5.2. Mellin Jackson kernels. For y ∈ R + , β ∈ N , γ ≥ , the Mellin Jackson kernels aredefined by J − γ,β ( y ) := d γ,β y − c sinc β (cid:18) log y γβπ (cid:19) , where d − γ,β := (cid:90) ∞ sinc β (cid:18) log y γβπ (cid:19) duu . The Mellin Jackson kernels also satisfy the conditions (i),(ii) and (iii) (see [6]). We canstudy the convergence of the I χw f with jump discontinuity associated with these kernelsonly for the case given in Theorem 2 and we observe that χ ( u ) (cid:54) = 0 , for every u ∈ (1 , e ).Hence, Theorem 3 and Theorem 5 can not be applied for these kernels. In order to findexamples of the kernels for which the exponential sampling series converge at any jumpdiscontinuity t ∈ R + of the given bounded signal f : R + → R , we need to constructsuitable kernels. One such construction is given below. Theorem 8. Let a, b ∈ R + and let χ a , χ b be two continuous kernels satisfying conditions(i), (ii) and (iii) such that supp χ a ⊆ [ e − a , e a ] and supp χ b ⊆ [ e − b , e b ] . For a fixed α ∈ R , define χ : R + → R by χ ( u ) := (1 − α ) χ a ( ue − a − ) + αχ b ( ue b ) , u ∈ R + . Then χ also satisfies conditions (i), (ii) and (iii) and in addition χ ( u ) = 0 , for every u ∈ [1 , e ) . Further, the Kantorovich exponential sampling series I χw f, w > correspondingto χ satisfying (i) of Theorem 5 at any jump discontinuity t ∈ R + of the given boundedmeasurable signal f with α as a parameter.Proof. We estimate the Mellin transform of χ ( u ) as follows: (cid:100) [ χ ] M ( s ) = (cid:90) ∞ (1 − α ) χ a ( te − a − ) t c + iv − dt + (cid:90) ∞ αχ b ( te b ) t c + iv − dt := I + I . Making substitution u = e − a − , du = e − a − dt in I and u = te b , du = e b dt in I , weeasily obtain (cid:100) [ χ ] M ( s ) = (1 − α ) (cid:92) [ χ a ] M ( s ) e (1+ a ) s + α (cid:91) [ χ b ] M ( s ) e − bs , where s = c + iv. It is easy to check that χ satisfies conditions (i) and (iii). Now we showthat kernel satisfies the condition (ii). We obtain (cid:100) [ χ ] M (2 kπi ) = (1 − α ) (cid:92) [ χ a ] M (2 kπi ) e (1+ a )(2 kπi ) + α (cid:91) [ χ b ] M (2 kπi ) e − b (2 kπi ) . Since χ a and χ b satisfies condition (ii), 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 = 0 . For suitable choice of a and b , we obtain (cid:100) [ χ ] M (2 kπi ) = (cid:26) , if k (cid:54) = 01 , if k = 0 . Hence, χ satisfies condition (ii) and we easily see that χ also satisfies χ ( u ) = 0 for1 ≤ u < e. Now, we obtain (cid:90) ∞ χ ( u ) u kπi duu = (1 − α ) (cid:90) ∞ χ a ( ue − a − ) u kπi duu = (1 − α ) e kπi (1+ a ) (cid:92) [ χ a ] M (2 kπi )= (cid:26) , if k (cid:54) = 01 − α, if k = 0 . Therefore, the condition (v) of Theorem is satisfied, hence the proof is completed. (cid:3) As a consequence of the following theorem, condition on the kernels can be given sothat M ν ( χ ) < + ∞ for some 0 ≤ ν < q, for some q < M ν ( χ ) = + ∞ for q < ν ≤ . Theorem 9. Let χ : R + → R be a function such that C | log u | p ≤ χ ( u ) ≤ C | log u | p , ∀ | log u | > M holds for suitable constants < C ≤ C and for some < p ≤ , and M > . Then, wehave M ν ( χ ) = (cid:26) + ∞ , p − ≤ ν ≤ < + ∞ , ≤ ν < p − . Proof. Let p − ≤ ν ≤ , and u ∈ R + be fixed. Since p − ν ≤ , we have M ν ( χ ) ≥ k =+ ∞ (cid:88) k = −∞ | χ ( e − k u ) || log u − k | ν ≥ k =+ ∞ (cid:88) k = −∞ | χ ( e − k u ) || log u − k | ν | log u − k | p | log u − k | p ≥ C (cid:88) | log u − k | >M | log u − k | ν − p = + ∞ . Let 0 ≤ ν < p − u ∈ R + . Since M ( χ ) < + ∞ and χ ( u ) = O ( | log u | − p ), as | u | → + ∞ with p > (cid:88) | log u − k | >M | log u − k | ν − p with p − ν > 1, is uniformly convergent for every u ∈ R + , we obtain k =+ ∞ (cid:88) k = −∞ | χ ( e − k u ) || log u − k | ν ≤ (cid:88) | log u − k |≤ M + (cid:88) | log u − k | >M | χ ( e − k u ) || log u − k | ν ≤ M ν M ( χ ) + C (cid:88) | log u − k | >M | log u − k | ν − p < + ∞ . Hence, the proof is completed. (cid:3) Implementation Results. We show the approximation of discontinuous function f ( t ) = 21 + et , < t < e , e ≤ t < e , e ≤ t < t , t ≥ I χw f at jump discontinuities at t = 1 e , t = e and t = 4 . We define χ c ( t ) = 25 ¯ B ( te − ) + 35 ¯ B ( te ) , where the kernel ¯ B is defined by¯ B ( x ) = − log x, < x < e x, e < x < , otherwise.It is easy to verify that χ c ( t ) satisfies the conditions (i), (ii), (iii) and χ c ( u ) = 0 , for every u ∈ [1 , e ) . Further we observe that the condition (i) of Theorem 5 is also satisfied with α = 35 . In view of Theorem 5 and Theorem 8, we have I χ c w f → f ( t + 0) + 25 f ( t − 0) as w → ∞ at a jump discontinuity points t == 1 e , t = e and t = 4 . Next we consider discontinuous kernels: χ d ( t ) := χ c ( t ) + θ ( t ) , t ∈ R + , where χ c ( t ) = 25 ¯ B ( te − ) + 35 ¯ B ( te ) . and θ ( t ) is defined by θ ( t ) = , t = e, e − , t = e , e , otherwise.Then, we observe that (cid:88) k ∈ Z θ ( e − k u ) = 0 , and ψ − χ ( u ) = 0 for every u ∈ [1 , e ) . Also weobserve that χ d ( u ) is not necessarily a continuous (see Fig.4) and it satisfies the condition(i), (ii), (iii) and χ d ( u ) = 0 , for every u ∈ [1 , e ) . Again from Theorem 5 and Theorem 8,we see that I χ d w f converges to 35 f ( t + 0) + 25 f ( t − 0) as w → ∞ at a jump discontinuitypoints t == 1 e , t = e and t = 4 . The convergence of I χ c w f and I χ d w f at discontinuity points t = 1 e , t = e and t = 4 of the function f has been tested and numerical results have beenprovided in Tables 1, 2 and 3. Figure 1. Plot of the kernel χ c ( t ) = 25 ¯ B ( te − ) + 35 ¯ B ( te ) . Figure 2. Approximation of f ( t ) by I χw f based on χ c ( t ) = 25 ¯ B ( te − ) +35 ¯ B ( te ) for w = 5 . Figure 3. Approximation of f ( t ) by I χw f based on χ c ( t ) = 25 ¯ B ( te − ) +35 ¯ B ( te ) for w = 10 . Table 1. Approximation of f at the jump discontinuity point t = 1 e by the Kantorovichexponential sampling series I χ c w f and I χ d w f based on χ c ( t ) , χ d ( t ) for different values of w > . The theoretical limit is: lim w →∞ ( I χ c w f ) (cid:18) e (cid:19) = lim w →∞ ( I χ d w f ) (cid:18) e (cid:19) = 35 f (cid:18) e + 0 (cid:19) + 25 f (cid:18) e − (cid:19) = 1 . .w I χ c w f . . . . . . . . . . I χ d w f . − . . − . . . . . . . Table 2. Approximation of f at the jump discontinuity point t = e by the Kantorovichexponential sampling series I χ c w f and I χ d w f based on χ c ( t ) , χ d ( t ) for different values of w > . The theoretical limit is: lim w →∞ ( I χ c w f )( e ) = lim w →∞ ( I χ d w f )( e ) = 35 f ( e + 0) + 25 f ( e − 0) = 2 . .w I χ c w f . . . . . . 600 2 . 600 2 . 600 2 . 600 2 . I χ d w f . . . . . . 600 2 . 600 2 . 600 2 . 600 2 . Table 3. Approximation of f at the jump discontinuity point t = 4 by the Kantorovichexponential sampling series I χ c w f and I χ d w f based on χ c ( t ) , χ d ( t ) for different values of w > . The theoretical limit is: lim w →∞ ( I χ c w f )(4) = lim w →∞ ( I χ d w f )(4) = 35 f (4 + 0) + 25 f (4 − 0) = 1 . .w I χ c w f . . . . . . . . . . I χ d w f . . . . . . . . . . Figure 4. Plot of the kernel χ d ( t ) = χ c ( t ) + θ ( t ) . Figure 5. Approximation of f ( t ) by I χw f based on χ d ( t ) = χ c ( t ) + θ ( t ) for w = 10 . Acknowledgments. The first and second author is supported by DST-SERB, IndiaResearch Grant EEQ/2017/000201. The third author P. Devaraj has been supported byDST-SERB Research Grant MTR/2018/000559. References [1] C. Bardaro, I. 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