Approximation by Durrmeyer type Exponential Sampling Series
aa r X i v : . [ m a t h . F A ] A ug Approximation by Durrmeyer type ExponentialSampling Series
Shivam Bajpeyi ∗ A. Sathish Kumar † Abstract
In this article, we analyze the approximation properties of the new family of Durrmeyer typeexponential sampling operators. We derive the point-wise and uniform approximation theoremand Voronovskaya type theorem for these generalized family of operators. Further, we constructa convex type linear combination of these operators and establish the better approximation re-sults. Finally, we provide few examples of the kernel functions to which the presented theorycan be applied along with the graphical representation.
Keywords.
Durrmeyer type exponential sampling operators. Point-wise convergence. Logarith-mic modulus of continuity. Mellin transform.
The sampling theory provides the prominent tool to handle various problems arising in signalprocessing and approximation theory. The classical sampling theorem is credit to the names ofWhittaker-Kotel’nikov-Shannon, which provides a reconstruction formula for the band-limitedfunctions using its ordinates at a series of points (see [5, 44]). In order to weaken the assumption,Brown [23], Butzer and Splettsst¨ober [24] initiated the study and since then, many authors havecontributed significantly in the development of the theory in various settings, see eg. [27, 6, 8,33, 25, 46, 32, 7, 2].The theory of exponential sampling began due to a group of engineers and physicists Pike,Bertero [22] and Gori [35] who introduced the exponential sampling formula (see also [43, 29]) ∗ Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, Nagpur-440010, India.E-mail: [email protected] † Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, Nagpur-440010, India.E-mail: [email protected] hich provides a series representation for the Mellin band limited functions exploiting its ex-ponentially spaced sample values. The theory of exponential sampling has been proven to bean important tool to deal with various inverse problems in the area of optical physics, see eg.[22, 31, 35, 43]. The mathematical study of the exponential sampling theorem was firstly initi-ated by Butzer and Jansche in [29] using the Mellin theory. The Mellin theory revealed to be themost suitable frame to put the theory of exponential sampling and found many applications inthe boundary value problems (see [29]). Mamedov pioneered the separate study of Mellin theoryindependent from Fourier analysis and investigated the approximation properties of Mellin con-volution operators in [41]. We mention here some of the significant developments in the Mellinanalysis [28, 29, 30, 9, 13, 17].Bardaro et.al.[18] made a remarkable development in this direction by generalizing the ex-ponential sampling formula, where they considered the generalized kernel with suitable assump-tions in place of lin c -function. This furnished a mechanism to approximate not necessarilyMellin band-limited functions using the sample values at nodes ( e kw ) w > , k ∈ Z . These operatorshave been studied in different settings in [12, 20, 21, 4]. In order to reduce the time-jitter error,the Kantorovich modification of these operators has been introduced and studied in [38, 45].The time-jitter error causes when the sample values can not be obtained exactly at the nodes.The Kantorovich type operators are known to reduce the time-jitter error as they calculate theinformation around the nodes rather than exactly at the nodes. This modification enables to ap-proximate integrable functions using the sample values which are exponentially spaced. TheKantorovich version of the several operators have been investigated extensively in various set-tings, see [6, 7, 42, 2, 46, 32, 40, 39, 1, 33]. The Kantorovich type exponential sampling serieswas defined in [38], namely ( I χ w f )( x ) = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw f ( e u ) du , w > f : R + → R is locally integrable such that the above series is convergent for every x ∈ R + .In view of Theorem 3.1 in [45], it is apparent that the Kantorovich exponential samplingoperators (1.1) fails to improve the order of approximation in the asymptotic formula. Thismotivates us to adopt another approach, known as Durrmeyer method, where the integral meanis replaced by a general convolution operator. This method was firstly applied to the Bernsteinpolynomials in the series of papers [34, 36, 37]. The Durrmeyer type modification is known toprovide the better order of approximation in various settings, see eg.[1, 14, 15, 19]. In this paper,we generalize the operator (1.1) by replacing the integral mean with the Mellin singular integralto obtain the following family of operators. For x ∈ R + and w > , we introduce the Durrmeyertype exponential sampling series as ( I χ , φ w f )( x ) = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w ) f ( t ) dtt . where f : R + → R is integrable such that the above series is convergent for every x ∈ R + .The paper is organized as follows. We begin with some auxiliary definitions in Section 3. InSection 4, we establish the basic convergence theorem and asymptotic formula for the proposedfamily operators (3.2). Using Peetre’s K functional [3, 45], we obtain the quantitative estimates f the Voronovskaya type theorem. Section 5 is devoted to the study of suitable linear combi-nations of the family of operators (3.2) which produce the better order of approximation. Theidea of considering the linear combinations of the operators is a prominent method to improvethe order of convergence, see [10, 11, 4, 14, 15, 45]. In last section, we discuss few examples ofthe well known kernels along with the graphical representations. Let R + be the multiplicative topological group equipped with the Haar measure µ ( S ) = Z S dtt , where dt represents the Lebesgue measure and S is any measurable set. We define L p ( µ , R ) = : L p ( µ ) , ≤ p < + ∞ , the Lebesgue spaces with respect to the measure µ , endowed with the usual p - norm. We consider M ( µ ) as the class of all measurable functions and L ∞ ( µ ) as the space ofall bounded functions with respect to the measure µ . Let X c denotes the space of all functions f : R + → R such that f ( x ) x c ∈ L ( µ ) for some c ∈ R , endowed with the norm k f k X c = Z + ∞ | f ( u ) | u c duu . The Mellin transform of f ∈ X c is defined byˆ M [ f ]( s ) : = Z + ∞ f ( u ) u s duu , ( s = c + it , t ∈ R ) . Indeed, if the Mellin transform exists for some c then it exists for every s = c + it . The basicproperties of the Mellin transform can be found in [28, 18]. Furthermore, the pointwise derivativein the Mellin’s frame is given by the following limit:lim h → τ ch f ( x ) − f ( x ) h − = x f ′ ( x ) + c f ( c ) , provided f ′ exists. Here, τ ch represents the Mellin translation operator and is defined as ( τ ch f )( x ) : = h c f ( hx ) . Thus the pointwise Mellin’s derivative θ c f of any function f : R + → C defined as ( θ c f )( x ) : = x f ′ ( x ) + c f ( x ) , x ∈ R + provided f ′ exists. Subsequently, the r th order Mellin differential operator can be expressed as θ rc : = θ c ( θ r − c ) . For the sake of convenience, we define θ c : = θ c and θ f : = θ f . Moreover, C ( R + ) denotes the space of all continuous and bounded functions on R + equippedwith the supremum norm k f k ∞ : = sup x ∈ R + | f ( x ) | . Subsequently, for any r ∈ N , C ( r ) ( R + ) be thesubspace of C ( R + ) such that f ( k ) , k ∈ N exists for every k ≤ r and each f ( k ) ∈ C ( R + ) . In whatfollows, we call a function f : R + → C log-uniformly continuous on R + if for any given ε > , there exists δ > | f ( u ) − f ( v ) | < ε whenever | log u − log v | < δ , for any u , v ∈ R + . We denote C ( R + ) the space of C ( R + ) containing all log-uniformly continuous and boundedfunctions defined on R + . Similarly, C ( r ) ( R + ) denotes the space of functions which are r -timescontinuously Mellin differentiable and θ r f ∈ C ( R + ) . The notion of log-continuity was firstintroduced in [41]. Durrmeyer type generalization of exponential samplingseries
Let χ : R + → R be the kernel which is continuous on R + . For any ν ∈ N : = N ∪ { } and x ∈ R + , we define its algebraic and absolute moments of order ν respectively as m ν ( χ , u ) : = + ∞ ∑ k = − ∞ χ ( e − k u )( k − log u ) ν , M ν ( χ ) : = sup u ∈ R + + ∞ ∑ k = − ∞ | χ ( e − k u ) || k − log u | ν . Moreover, let φ be the kernel such that φ ∈ L ( µ ) . Then the algebraic and absolute moments forthe kernel φ are defined byˆ m ν ( φ ) : = Z ∞ φ ( u ) log ν u duu , ˆ M ν ( φ ) : = Z ∞ | φ ( u ) | | log u | ν duu . We suppose that the kernel functions χ and φ satisfy the following conditions:K1) For every u ∈ R + , + ∞ ∑ k = − ∞ χ ( e − k u ) = Z ∞ φ ( u ) duu = . K2) For some r ∈ N , ( M r ( χ ) + ˆ M r ( φ )) < + ∞ andlim γ → + ∞ ∑ | k − log u | > γ | χ ( e − k u ) | | k − log u | r = u ∈ R + . Remark 3.1 [38] It is easy that for µ , ν ∈ N with µ < ν , M ν ( χ ) < + ∞ implies that M µ ( χ ) < + ∞ . Moreover, the condition (K2) implies that there holds lim γ → + ∞ ∑ | k − log u | > γ | χ ( e − k u ) | | k − log u | j = for j = , , ..., r − . Following along the lines of Remark 4.1 in [38], we deduce that ˆ M ν ( φ ) < + ∞ implies thatˆ M µ ( φ ) < + ∞ whenever µ < ν . Under the above assumptions on the kernels χ and φ , we define the Durrmeyer type gener-alized exponential sampling series as ( I χ , φ w f )( x ) = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w ) f ( t ) dtt , w > f : R + → R is integrable such that the above series is convergent for every x ∈ R + . It isevident that for f ∈ L ∞ ( µ ) , the series (3.2) is well-defined, that is, L ∞ ( µ ) ∈ Dom ( I χ , φ w ) , where Dom ( I χ , φ w ) consists of all functions f : R + → R such that the series (3.2) is absolutely convergentfor any x ∈ R + . emark 3.2 Let χ and φ be the kernel functions. Then the Mellin convolution integral of f with φ w ( u ) : = w φ ( u w ) is given by ( T φ w f )( s ) : = (cid:0) f ∗ φ w ( u ) (cid:1) ( s ) = Z ∞ f ( t ) φ w (cid:16) ts (cid:17) dtt = w Z ∞ f ( t ) φ (cid:18) t w s w (cid:19) . Now we can represent our proposed operator (3.2) in terms of the generalized exponential sam-pling operator ( S χ w ) w > introduced by Bardaro et.al. in [18], using the Mellin convolution inte-gral as follows ( I χ , φ w f )( x ) : = ( S χ w (cid:0) T φ w f (cid:1) )( x ) , x ∈ R + . Remark 3.3
For any y ∈ R + , we define φ ( y ) : = κ [ , e ] ( y ) = ( , ≤ y < e , otherwisewhere κ represents the characteristic function. Considering the change of variable t = e u , weobtain φ ( e wu − k ) = ( , kw ≤ u ≤ k + w , otherwise . Then, the corresponding generalized operator (3.2) acquires the following form: ( I χ w f )( x ) = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw f ( e u ) du . (3.3) This particular family of operators (3.3) was analyzed in [38, 45].
The purpose of this section is to derive some local approximation results for the Durrmeyer typeexponential sampling operators (3.2).
Theorem 4.1
Let f ∈ M ( µ ) ∩ L ∞ ( µ ) . Then the series (3.2) converges to f ( x ) at every pointx ∈ R + , the point of continuity of f . Moreover, for f ∈ C ( R + ) , we have lim w → ∞ k I χ , φ w f − f k ∞ = . Proof.
Using the condition (K1), we can write | ( I χ , φ w f )( x ) − f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w )( f ( t ) − f ( x )) dtt (cid:12)(cid:12)(cid:12)(cid:12) ≤ + ∞ ∑ k = − ∞ (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) w (cid:18) Z | log t − log x | < δ + Z | log t − log x |≥ δ (cid:19) (cid:12)(cid:12) φ ( e − k t w ) (cid:12)(cid:12) (cid:12)(cid:12) f ( t ) − f ( x ) (cid:12)(cid:12) dtt : = I + I . et x ∈ R + be the point of continuity of f then for any fixed ε > δ > | f ( t ) − f ( x ) | < ε , whenever | log t − log x | < δ . Now, considering the change of variable e − k t w = : p , we obtain | I | ≤ + ∞ ∑ k = − ∞ (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) ε Z w ( log x + δ ) − kw ( log x − δ ) − k (cid:12)(cid:12) φ ( p ) (cid:12)(cid:12) d pp ≤ ε (cid:0) M ( χ ) k φ k (cid:1) . Similarly, we estimate I . | I | ≤ k f k ∞ ∑ k ∈ Z (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) (cid:18) Z w ( log x − δ ) − k − ∞ + Z + ∞ w ( log x + δ ) − k (cid:19) | φ ( p ) | d pp = : I ′ + I ′′ . First we consider I ′ . Using the fact that φ ∈ L ( µ ) and condition (K2), for a fixed x , δ we obtain | I ′ | ≤ k f k ∞ ∑ | w log x − k | < w δ + ∑ | w log x − k |≥ w δ (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) Z w ( log x − δ ) − k − ∞ | φ ( p ) | d pp ≤ k f k ∞ ε ( + k φ k ) . Similarly, we deduce that | I ′′ | ≤ k f k ∞ ε ( + k φ k ) . Finally combining the estimates I − I , weget the desired result. Subsequently for f ∈ C ( R + ) , the proof follows in the similar manner.Our next result is the following asymptotic formula for the family of operators ( I χ , φ w ) w > . Theorem 4.2
Let f ∈ C ( r ) ( R + ) locally at x ∈ R + and χ , φ be the kernels. Then we have (cid:2) ( I χ , φ w f )( x ) − f ( x ) (cid:3) = r ∑ j = θ j f ( x ) j ! w j j ∑ η = (cid:18) j η (cid:19) ˆ m j − η ( φ ) m η ( χ , x ) ! + R w , r ( x ) , where R w , r ( x ) = ∑ k ∈ Z χ ( e − k x w ) w Z ∞ φ ( e − k t w ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt . Moreover R w , r ( x ) = o ( w − r ) as w → + ∞ for every x ∈ R + . Proof.
Using Taylor’s formula in terms of Millan derivatives (see [18, 38]), we write f ( t ) − f ( x ) = r ∑ j = θ j f ( x ) j ! ( log t − log x ) j + h (cid:16) tx (cid:17) ( log t − log x ) r , where h is a bounded function such that lim y → h ( y ) = . In view of (3.2) we obtain [( I χ , φ w f )( x ) − f ( x )]= + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w ) " r ∑ j = θ j f ( x ) j ! ( log t − log x ) j + h (cid:16) tx (cid:17) ( log t − log x ) r dtt = r ∑ j = θ j f ( x ) j ! + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w )( log t − log x ) j + R w , r ( x ) , here R w , r ( x ) = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt . For any fixed index ξ ∈ N , we have + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w ) θ ξ f ( x ) ξ ! ( log t − log x ) ξ dtt = θ ξ f ( x ) ξ ! + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w )( log t − log x ) ξ dtt = θ ξ f ( x ) ξ ! w ξ + ∞ ∑ k = − ∞ χ ( e − k x w ) Z ∞ φ ( p )( log p + k − w log x ) ξ d pp = θ ξ f ( x ) ξ ! w ξ + ∞ ∑ k = − ∞ χ ( e − k x w ) Z ∞ φ ( p ) ξ ∑ η = (cid:18) ξη (cid:19) ( log p ) ( ξ − η ) ( k − w log x ) η ! d pp = θ ξ f ( x ) ξ ! w ξ ξ ∑ η = (cid:18) ξη (cid:19) ˆ m ξ − η ( φ ) m η ( χ , x ) ! . Thus we obtain [( I χ , φ w f )( x ) − f ( x )] = r ∑ j = θ j f ( x ) j ! w j j ∑ η = (cid:18) j η (cid:19) ˆ m j − η ( φ ) m η ( χ , x ) ! + R w , r ( x ) . (4.4)Now we estimate the remainder term R w , r ( x ) in (4.4). In order to do that let ε > δ > | h ( y ) | < ε whenever | log y | < δ . We write R w , r ( x ) as | R w , r | ≤ ∑ k ∈ Z (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w Z | log t − log x | < δ φ ( e − k t w ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∑ k ∈ Z (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w Z | log t − log x |≥ δ φ ( e − k t w ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : = I ′ + I ′′ . Thus we have | I ′ | ≤ ε w r r ∑ j = (cid:18) rj (cid:19) M ( r − j ) ( χ ) ˆ M j ( φ ) ! . Using the fact that h is bounded, we obtain | I ′′ | ≤ k h k ∞ ∑ k ∈ Z (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w Z | log t − log x |≥ δ φ ( e − k t w )( log t − log x ) r dtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k h k ∞ w r ∑ k ∈ Z (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) Z | log p + k − w log x |≥ w δ | φ ( p ) || log p + k − w log x | r d pp ≤ r − ε k h k ∞ w r ( M r ( χ ) + k φ k ) . n Combining the estimates of I ′ − I ′′ , we conclude that R w , r ( x ) = o ( w r ) as w → + ∞ and hencethe desired result is established.From the above theorem, we deduce the following Voronovskaja type asymptotic result. Corollary 4.1
Let f ∈ C ( ) ( R + ) and χ , φ be the kernels. Then for every x ∈ R + , lim w → ∞ w (cid:2) ( I χ , φ w f )( x ) − f ( x ) (cid:3) = θ f ( x ) ( m ( χ , x ) + ˆ m ( φ )) . Corollary 4.2
Under the assumptions of Theorem 4.2, if in addition, the kernels χ , φ satisfym j ( χ , x ) = and ˆ m j ( φ ) = for j = , , ... r − , then the following holds lim w → ∞ w r [( I χ , φ w f )( x ) − f ( x )] = θ r f ( x ) r ! ( m r ( χ , x ) + ˆ m r ( φ )) . Remark 4.1
The condition that f is bounded, can be relaxed by assuming that | f ( e x ) | ≤ ( a + b | x | ) , x ∈ R + , where a , b ∈ R are arbitrary constants. Indeed, from (3.2), we have | ( I χ , φ w f )( x ) | ≤ + ∞ ∑ k = − ∞ | χ ( e − k x w ) | w Z ∞ | φ ( e − k t w ) | ( a + b | log t | ) dtt ≤ + ∞ ∑ k = − ∞ | χ ( e − k x w ) | Z ∞ | φ ( p ) | (cid:18) a + bw (cid:12)(cid:12) log p + k (cid:12)(cid:12)(cid:19) d pp ≤ ( a + b | log x | ) M ( χ ) ˆ M ( φ ) + bw (cid:0) M ( χ ) ˆ M ( φ ) + M ( χ ) ˆ M ( φ ) (cid:1) . Under the assumptions on the kernels χ and φ , we conclude that f ∈ Dom ( I χ , φ w ) . We establish the quantitative estimate of the asymptotic formula derived in Theorem 4.2, in termsof Peetre’s K-functional [3, 45]. The Peetre’s K-functional for f ∈ C ( r ) ( R + ) is defined byˆ K ( f , ε , C ( r ) ( R + ) , C ( r + ) ( R + )) : = inf {k θ r ( f − g ) k ∞ + ε k θ r + g k ∞ : g ∈ C ( ) ( R + ) , ε ≥ } . Theorem 4.3
Let χ , φ be the kernel functions and f ∈ C ( r ) ( R + ) . Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [( I χ w f )( x ) − f ( x )] − r ∑ j = θ j f ( x ) j ! w j j ∑ η = (cid:18) j η (cid:19) ˆ m j − η ( φ ) m η ( χ , x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ r Ar ! w r ˆ K (cid:18) θ r f , ( r + ) w BA (cid:19) . where A : = (cid:0) M ( χ ) ˆ M r ( φ ) + M r ( χ ) ˆ M ( φ ) (cid:1) and B : = (cid:0) M ( χ ) ˆ M r + ( φ ) + M r + ( χ ) ˆ M ( φ ) (cid:1) . Proof.
From Theorem 3.2, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [( I χ w f )( x ) − f ( x )] − r ∑ j = θ j f ( x ) j ! w j j ∑ η = (cid:18) j η (cid:19) ˆ m j − η ( φ ) m η ( χ , x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z ∞ φ ( e − k t w ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ow we substitute R r ( f , x , t ) : = h (cid:16) tx (cid:17) ( log t − log x ) r . Then by using the following estimate(see [13, 45]) | R r ( f , x , t ) | ≤ r ! | log t − log x | r ˆ K (cid:18) θ r f , | log t − log x | ( r + ) (cid:19) , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [( I χ w f )( x ) − f ( x )] − r ∑ j = θ j f ( x ) j ! w j j ∑ η = (cid:18) j η (cid:19) ˆ m j − η ( φ ) m η ( χ , x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ + ∞ ∑ k = − ∞ | χ ( e − k x w ) | w Z ∞ | φ ( e − k t w ) | (cid:18) r ! | log t − log x | r (cid:19) ˆ K (cid:18) θ r f , | log t − log x | ( r + ) (cid:19) dtt ≤ + ∞ ∑ k = − ∞ | χ ( e − k x w ) | ( w ) Z ∞ | φ ( e − k t w ) | | log t − log x | r r ! (cid:18) k θ r ( f − g ) k ∞ + | log t − log x | ( r + ) k θ r + g k ∞ (cid:19) dtt ≤ k θ r ( f − g ) k ∞ r ! + ∞ ∑ k = − ∞ | χ ( e − k x w ) | w Z ∞ | φ ( e − k t w ) | | log t − log x | r dtt + k θ r + g k ∞ ( r + ) ! + ∞ ∑ k = − ∞ | χ ( e − k x w ) | w Z ∞ | φ ( e − k t w ) | | log t − log x | r + dtt : = I + I . First we evaluate I . | I | ≤ k θ r ( f − g ) k ∞ r ! w r + ∞ ∑ k = − ∞ | χ ( e − k x w ) | Z ∞ | φ ( p ) || log p + k − w log x | r d pp ≤ r k θ r ( f − g ) k ∞ r ! w r + ∞ ∑ k = − ∞ | χ ( e − k x w ) | Z ∞ | φ ( p ) || log p | r d pp ! + r k θ r ( f − g ) k ∞ r ! w r + ∞ ∑ k = − ∞ | χ ( e − k u ) || k − w log x | r Z ∞ | φ ( p ) | d pp ! ≤ r k θ r ( f − g ) k ∞ r ! w r (cid:0) M ( χ ) ˆ M r ( φ ) + M r ( χ ) ˆ M ( φ ) (cid:1) . Similarly, we have | I | ≤ k θ r + g k ∞ ( r + ) ! w r + + ∞ ∑ k = − ∞ | χ ( e − k x w ) | Z ∞ | φ ( p ) | | log p + k − w log x | r + d pp ≤ r k θ r + g k ∞ ( r + ) ! w r + (cid:0) M ( χ ) ˆ M r + ( φ ) + M r + ( χ ) ˆ M ( φ ) (cid:1) . On combining the estimates I − I and taking the infimum to g ∈ C ( r + ) ( R + ) , we establish theproof. As a consequence of the above result, we deduce the following corollary. Corollary 4.3
Let χ , φ be the kernel functions and f ∈ C ( ) ( R + ) . Then the following estimateholds (cid:12)(cid:12)(cid:12) [( I χ w f )( x ) − f ( x )] − ( θ f )( x ) (cid:0) ˆ m ( φ ) + m ( χ ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ Aw ˆ K (cid:18) f , w BA (cid:19) , here A : = (cid:0) M ( χ ) ˆ M ( φ ) + M ( χ ) ˆ M ( φ ) (cid:1) and B : = (cid:0) M ( χ ) ˆ M ( φ ) + M ( χ ) ˆ M ( φ ) (cid:1) . The aim of this section is to formulate the combinations of the family of operators (3.2) toproduce better order of convergence in the asymptotic formula for f ∈ C ( r ) ( R + ) without usingthe constraint that the higher order algebraic moments for the kernels χ , φ vanish on R + . Weconstruct the linear combination of the family ( I χ , φ w ( f , . )) w > in the following manner.Let β i , i = , , ..., p be non-zero real numbers such that p ∑ i = β i = . For x ∈ R + and w > , we define the linear combination of the operators (3.2) as ( I χ , φ w , p )( f , x ) = p ∑ i = β i + ∞ ∑ k = − ∞ χ ( e − k x iw ) ( iw ) Z ∞ φ ( e − k t iw ) f ( t ) dtt = p ∑ i = β i ( I χ , φ iw )( f , x ) . (5.5)Now we derive the asymptotic formula for the family of operators defined in (5.5). Theorem 5.1
Let χ , φ be the kernels and f ∈ C ( r ) ( R + ) locally at x ∈ R + . Then we have [( I χ , φ w , p )( f , x ) − f ( x )] = r ∑ j = ( θ j f )( x ) j ! w j M pj ( χ , φ ) + o ( w − r ) , where M pj ( χ , φ ) : = p ∑ i = β i i j j ∑ η = (cid:18) j η (cid:19) ˆ m j − η ( φ ) m η ( χ , x ) ! . Proof.
From the condition p ∑ i = β i = , we can write [( I χ w , p )( f , x ) − f ( x )] = p ∑ i = β i + ∞ ∑ k = − ∞ χ ( e − k x iw ) ( iw ) Z ∞ φ ( e − k t w )( f ( t ) − f ( x )) dtt . Now using the r th order Mellin’s Taylor formula, we obtain [( I χ w , p )( f , x ) − f ( x )]= p ∑ i = β i + ∞ ∑ k = − ∞ χ ( e − k x iw ) ( iw ) Z ∞ φ ( e − k t iw ) r ∑ j = ( θ j f )( x ) j ! ( log t − log x ) j + h (cid:16) tx (cid:17) ( log t − log x ) r ! dtt = p ∑ i = β i + ∞ ∑ k = − ∞ χ ( e − k x iw ) ( iw ) Z ∞ φ ( e − k t iw ) r ∑ j = ( θ j f )( x ) j ! ( log t − log x ) j dtt ! + p ∑ i = β i + ∞ ∑ k = − ∞ χ ( e − k x iw ) ( iw ) Z ∞ φ ( e − k t iw ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt = p ∑ i = β i r ∑ j = ( θ j f )( x ) j ! w j i j j ∑ η = o (cid:18) j η (cid:19) ( ˆ m j − η ( χ , x ) m j ( φ ) !! + R iw , r , here R iw , r = p ∑ i = β i + ∞ ∑ k = − ∞ χ ( e − k x iw ) ( iw ) Z ∞ φ ( e − k t iw ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt . Now proceedingalong the lines of Theorem 4.2, it follows that R iw , r = o ( w − r ) as w → ∞ . Hence, the proof iscompleted.From the above result, we establish the following Voronovskaja type asymptotic formula.
Corollary 5.1
For f ∈ C ( ) ( R + ) , we havew [( I χ , φ w , p )( f , x ) − f ( x )] = p ∑ i = β i i ( θ f ( x )( ˆ m ( φ ) + m ( χ , x ))) + o ( w − ) . Furthermore, if m j ( χ , x ) = = ˆ m j ( φ ) for ≤ j ≤ r − , then the following holdsw r [( I χ w , p )( f , x ) − f ( x )] = p ∑ i = β i i r (cid:18) ( θ r f )( x ) r ! ( m r ( χ , x ) + ˆ m r ( φ )) (cid:19) + o ( w − r ) . Corollary 5.2
Under the conditions of Theorem 5.2, if moreover ¯ M pk ( χ ) = , fork = , , · · · ( p − ) , then for f ∈ C ( r ) ( R + ) with r ≥ p , we have lim w → ∞ w p [( I χ w , p f )( x ) − f ( x )] = ( θ p f )( x )( p + ) ! ¯ M pp ( χ , φ ) . It is important to mention here that M pk ( χ , φ ) does not vanish, in general. In order to have M pk ( χ , φ ) = k = , , , · · · p − , we need to solve the following system p ∑ i = β i = , p ∑ i = β i i = , p ∑ i = β i i = , · · · p ∑ i = β i i p − = . (5.6)The solution of the above system yields a linear combination which provides the convergence oforder at least p for functions f ∈ C ( p ) ( R + ) . In particular, let f ∈ C ( r ) ( R + ) , r ≥ . Then it is evident from Theorem 4.2 that the operator(3.2) converges linearly, in general. In order to improve the rate of convergence, we need tosolve the system (5.6) for p = β = , β = − β = . Usingthese coefficient, we establish the following linear combination ( I χ , φ w , )( f , x ) = ( I χ , φ w )( f , x ) + ( − )( I χ , φ w )( f , x ) + ( I χ , φ w )( f , x ) (5.7)which corresponds to the asymptotic formula, for r ≥ , given by [( I χ , φ w , )( f , x ) − f ( x )] = r ∑ j = ( θ j f )( x )( j + ) ! w j M j ( χ , φ ) + o ( w − r ) . It can be observe that the above linear combination produces the rate of convergence atleast 3for f ∈ C ( r ) ( R + ) , r ≥ . .1 Quantitative estimates of linear combination Theorem 5.2
Let χ , φ be the kernels and f ∈ C ( r ) ( R + ) . Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [( I χ , φ w , p )( f , x ) − f ( x )] − r ∑ j = ( θ j f )( x ) j ! w j ¯ M pj ( χ , φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ r k θ r ( f − g ) k ∞ r ! w r p ∑ i = β i i r ˆ K (cid:18) θ r f , BA w ( r + ) (cid:19) , where A = p ∑ i = β i i r (cid:0) M ( χ ) ˆ M r ( φ ) + M r ( χ ) ˆ M ( φ ) (cid:1) and B = p ∑ i = β i i r + (cid:0) M ( χ ) ˆ M r + ( φ ) + M r + ( χ ) ˆ M ( φ ) (cid:1) . Proof.
In view of Theorem 5.1, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [( I χ w f )( x ) − f ( x )] − r ∑ j = ( θ j f )( x ) j ! w j ¯ M pj ( χ , φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∑ i = β i + ∞ ∑ k = − ∞ χ ( e − k x iw ) ( iw ) Z ∞ φ ( e − k t iw ) h (cid:16) tx (cid:17) ( log t − log x ) r dtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now we substitute R r ( f , x , t ) : = h (cid:16) tx (cid:17) ( log t − log x ) r . Then by using the estimate | R r ( f , x , t ) | ≤ r ! | log t − log x | r ˆ K (cid:18) θ r f , | log t − log x | r ( r + ) (cid:19) , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [( I χ w f )( x ) − f ( x )] − r ∑ j = θ j f ( x ) j ! w j j ∑ η = (cid:18) j η (cid:19) ˆ m j − η ( φ ) m η ( χ , x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p ∑ i = β i + ∞ ∑ k = − ∞ | χ ( e − k x iw ) | ( iw ) Z ∞ | φ ( e − k t iw ) | (cid:18) r ! | log t − log x | r (cid:19) ˆ K (cid:18) θ r f , | log t − log x | ( r + ) (cid:19) dtt ≤ k θ r ( f − g ) k ∞ r ! p ∑ i = β i + ∞ ∑ k = − ∞ | χ ( e − k x iw ) | ( iw ) Z ∞ | φ ( e − k t iw ) | | log t − log x | r dtt + k θ r + g k ∞ ( r + ) ! p ∑ i = β i + ∞ ∑ k = − ∞ | χ ( e − k x iw ) | ( iw ) Z ∞ | φ ( e − k t iw ) | | log t − log x | r + dtt : = I + I . irst we evaluate I . | I | ≤ k θ r ( f − g ) k ∞ r ! w r p ∑ i = β i i r + ∞ ∑ k = − ∞ | χ ( e − k x iw ) | Z ∞ | φ ( p ) || log p + k − iw log x | r d pp ≤ r k θ r ( f − g ) k ∞ r ! w r p ∑ i = β i i r + ∞ ∑ k = − ∞ | χ ( e − k x iw ) | Z ∞ | φ ( p ) || log p | r d pp + + ∞ ∑ k = − ∞ | χ ( e − k x iw ) || k − iw log x | r Z ∞ | φ ( p ) | d pp ! ≤ r k θ r ( f − g ) k ∞ r ! w r p ∑ i = β i i r (cid:0) M ( χ ) ˆ M r ( φ ) + M r ( χ ) ˆ M ( φ ) (cid:1) . Similarly, we obtain | I | ≤ k θ r + g k ∞ ( r + ) ! w r + p ∑ i = β i i r + + ∞ ∑ k = − ∞ | χ ( e − k x iw ) | Z ∞ | φ ( p ) | | log p + k − iw log x | r + d pp ≤ r k θ r + g k ∞ ( r + ) ! w r + p ∑ i = β i i r + (cid:0) M ( χ ) ˆ M r + ( φ ) + M r + ( χ ) ˆ M ( φ ) (cid:1) . Now using the estimates of I − I and then passing the infimum over g ∈ C ( r + ) ( R + ) , we getthe desired result. Corollary 5.3
Let χ , φ be the kernel functions and f ∈ C ( ) ( R + ) . Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [( I χ , φ w , p )( f , x ) − f ( x )] − ( θ f )( x ) w ¯ M p ( χ , φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k θ r ( f − g ) k ∞ w p ∑ i = β i i ˆ K (cid:18) θ f , BA w (cid:19) , where A = p ∑ i = β i i (cid:0) M ( χ ) ˆ M ( φ ) + M ( χ ) ˆ M ( φ ) (cid:1) and B = p ∑ i = β i i (cid:0) M ( χ ) ˆ M ( φ ) + M ( χ ) ˆ M ( φ ) (cid:1) . In this section, we present few examples of the kernels satisfying the assumptions of the dis-cussed theory along with the graphical representation. We begin with the well-known MellinB-spline kernel, see [18, 38, 45].
The Mellin B-splines of order n are defined as¯ B n ( x ) : = ( n − ) ! n ∑ j = ( − ) j (cid:18) nj (cid:19)(cid:18) n + log x − j (cid:19) n + + , x ∈ R + . he Mellin transformation of ¯ B n is given byˆ M [ ¯ B n ]( c + it ) = (cid:18) sin ( t )( t ) ! n , t = . (6.8)Since ¯ B n ( x ) is compactly supported for every n ∈ N , the condition (K2) is satisfied. For (K1),we use the following lemma derived in [38] using the Mellin Poisson summation formula [26],which has the following form ( i ) j + ∞ ∑ k = − ∞ χ ( e − k u )( k − log u ) j = + ∞ ∑ k = − ∞ ˆ M ( j ) [ χ ]( k π i ) u − k π i , for k ∈ Z . Lemma 6.1 [38] The condition + ∞ ∑ k = − ∞ χ ( e − k x w ) = is equivalent to the condition ˆ M [ χ ]( k π i ) = ( , if k = , if otherwiseMoreover, the condition m j ( χ , u ) = for j = , , ... n is equivalent to the condition ˆ M ( j ) [ χ ]( k π i ) = for j = , ..., n and ∀ k ∈ Z . In view of Lemma 6.1 and (6.8), it can be seen that ¯ B n ( x ) satisfies the condition (K1) for every n ∈ N and x ∈ R + . Now we consider χ ( x ) : = ¯ B ( x ) and φ ( x ) : = ˆ B ( x ) . Using the Mellin Poisson summationformula, we obtain the algebraic moments for χ ( x ) as m ( χ ) = , m ( χ ) = , m ( χ ) = , m ( χ ) = . Moreover, the algebraic moments for φ ( x ) are given byˆ m ( φ ) = , ˆ m ( φ ) = , ˆ m ( φ ) = , ˆ m ( φ ) = . In view of Theorem 4.2, the asymptotic formula for f ∈ C ( ) ( R + ) is given bylim w → ∞ w [ I χ , φ w f )( x ) − f ( x ] = ( θ f )( x ) . But, the linear combination (5.7) leads to the following asymptotic formulalim w → ∞ w [ I χ , φ w , f )( x ) − f ( x ] = , which ensures the order of convergence atleast 3 for f ∈ C ( ) ( R + ) . Next we compare the approximation by the family of operators (3.2) and the linear combi-nations of these operators using graphs and error estimates. It is apparent from
Figure 1,2 and
Table 1,2 that the linear combination of the proposed operators yield the better approximation. .5 4 4.5 5 5.5 6−30−20−100102030 X−axis Y − a x i s w=25w=45w=90f(x) Figure 1: This figure shows the approximation of f ( x ) = x cos ( π x ) , x ∈ ( π , π ) by ( I ¯ B , ¯ B w f )( x ) for w = , ,
90 respectively.
Table 1
Error estimation (upto decimal points) in the approximation of f ( x ) by ( I ¯ B , ¯ B w f )( x ) for w = , , . x | f ( x ) − ( I ¯ B , ¯ B f )( x ) | | f ( x ) − ( I ¯ B , ¯ B f )( x ) | | f ( x ) − ( I ¯ B , ¯ B f )( x ) | .
55 2 . . . .
98 4 . . . .
22 1 . . . .
85 4 . . . .
35 6 . . . Table 2
Error estimation (upto decimal points) in the approximation of g ( x ) by ( I ¯ B , ¯ B w g )( x ) and ( I ¯ B , ¯ B w , i g )( x ) for i = , and w = . x | g ( x ) − ( I ¯ B , ¯ B g )( x ) | | g ( x ) − ( I ¯ B , ¯ B , g )( x ) | | g ( x ) − ( I ¯ B , ¯ B , g )( x ) | .
75 0 . . . .
10 0 . . . .
85 0 . . . .
45 0 . . . .
95 0 . . . Y − a x i s I I I g(x) Figure 2: This figure exhibits the approximation of g ( x ) = x e − sin x , x ∈ (cid:0) π , (cid:1) by ( I ¯ B , ¯ B w g )( x ) and ( I ¯ B , ¯ B w , i g )( x ) for i = , w = Consider the linear combination of translates of B-spline functions of order n in the Mellinsetting as follows: ψ ( x ) : = c [ τ a ¯ B n ( x )] + c [ τ b ¯ B n ( x )]= c [ ¯ B n ( ax )] + c [ ¯ B n ( bx )] , ∀ x ∈ R + , a , b ∈ R . Using the linearity of Mellin-tranform, we obtainˆ M [ ψ ]( w ) = c ˆ M [ ¯ B n ( ax )] + c ˆ M [ ¯ B n ( bx )]= c a − w ˆ M [ ¯ B n ]( w ) + c b − w ˆ M [ ¯ B n ]( w ) . (6.9)On differentiating (6.9), we haveˆ M ′ [ ψ ]( w ) = c ( a − w ( ˆ M ′ [ ¯ B n ]( w ) − a − w log a )) + c ( b − w ( ˆ M ′ [ ¯ B n ]( w ) − b − w log b )) . In view of Lemma 6.1, we have the following system c + c = , c log a + c log b = . On solving for c and c , we get c = log b ( log b − log a ) , c = − log a ( log b − log a ) . n particular, for a = e − and b = e − , we obtain the following linear combination of the Mellin’sB-spline of order 2 ψ ( x ) : = B ( e − x ) − B ( e − x ) , x ∈ R + . Using Mellin Poisson summation formula, we obtain m ( ψ ) = ˆ M [ ψ ]( ) = m ( ψ ) = ˆ M ′ [ ψ ]( ) = . Similarly we calculate m ( ψ ) = , m ( ψ ) = − . Now for f ∈ C ( ) ( R + ) , the asymptotic formula is given aslim w → ∞ w [ I ψ , ˆ B w f )( x ) − f ( x ] = ( θ f )( x ) . But, the linear combination (5.7) leads to the following asymptotic formula:lim w → ∞ w [ I ψ , ˆ B w f )( x ) − f ( x ] = − ( θ f )( x ) . This shows that the combination (5.7) provides order of convergence atleast 3 for f ∈ C ( ) ( R + ) . References [1] Acu, M., Gupta,V., Tachev, G.; Better Numerical Approximation by Durrmeyer Type Op-erators, Results Math., 74-90 (2019).[2] Angeloni, L., Costarelli, D., Vinti, G.; A characterization of the absolute continuity interms of convergence in variation for the sampling Kantorovich operators, Med. J. Math.16:44 (2019).[3] Anastassiou, George A., Gal, Sorin G.; Approximation theory. Moduli of continuity andglobal smoothness preservation. Birkhauser Boston, Inc., Boston, MA, 2000.[4] Balsamo, S., Mantellini, I.; On linear combinations of general exponential sampling series,Results Math. 74 (180) (2019).[5] Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.; Approximation of the Whittaker SamplingSeries in terms of an Average Modulus of Smoothness covering Discontinuous Signals. J.Math. Anal. Appl. 316, 269-306(2006).[6] Bardaro, C., Vinti, G., Butzer, P.L., Stens, R.; Kantorovich-type generalized sampling se-ries in the setting of Orlicz spaces. Sampl. Theory Signal Image Process. 6, 29-52 (2007).[7] Bardaro, C., Mantellini, I.; Voronovskaya formulae for Kantorovich type generalized sam-pling series. Int. J. Pure Appl. Math. 62 , 247-262 (2010).[8] Bardaro, C., Butzer, P.L., Stens, R., Vinti, G.; Prediction by samples from the past witherror estimates covering discontinuous signals, IEEE Transactions on Information Theory.56(1), 614-633 (2010).
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