Direct and inverse results for Kantorovich type exponential sampling series
aa r X i v : . [ m a t h . NA ] F e b Direct and inverse results for Kantorovich typeexponential sampling series
Sathish Kumar Angamuthu ∗ Shivam Bajpeyi † Abstract
In this article, we analyze the behaviour of the new family of Kantorovich type exponentialsampling series. We obtain the point-wise approximation theorem and Voronovskaya type the-orem for the series ( I χ w ) w > . Further, we obtain a representation formula and an inverse resultapproximation for these operators. Finally, we give some examples of kernel functions to whichthe theory can be applied along with the graphical representation.
Keywords.
Kantorovich type exponential sampling series. Pointwise convergence. Logarithmicmodulus of continuity. Mellin transform. Inverse result.
The exponential sampling methods play the key role in solving problems in the area of opticalphysics and engineering, precisely in the phenomena like Fraunhofer diffraction, light scatteringetc [22, 21, 29, 36]. It all began when a group of optical physicists and engineers Bartero, Pike[21] and Gori [29] presented a representation formula known as exponential sampling formula ,for the class of Mellin band-limited function having exponentially spaced sample points, whichis also considered as the Mellin-version of the well known
Shannon sampling theorem ( see[6]). But, the pioneering idea of mathematical study of exponential sampling formula is creditedto Butzer and Jansche. Butzer et al. [9] proved the exponential sampling formula mathemat-ically using the theory of Mellin transform and Mellin approximation, which was first studied ∗ 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] eparately by Mamedov [33] and then developed by Butzer and Jansche [7, 8, 9, 10]. We men-tion some of the work related to the theory of Mellin transform and Mellin approximation (see[15, 17, 18]). Bardaro et al.[19] made a significant development in this direction when they re-placed the lin c function in the exponential sampling formula by more general kernel function.Let x ∈ R + and w > . Then, the generalized exponential sampling series (see [19]) is definedby ( S χ w f )( x ) = + ∞ ∑ k = − ∞ χ ( e − k x w ) f ( e kw ) (1.1)where f : R + → R be any function for which the series is absolutely convergent. The conver-gence of the above series studied in Mellin-Lebesgue spaces in [20]. The importance of the aboveseries appears when we need to reconstruct a not necessarily Mellin band-limited signal in ap-proximate sense having samples which are exponentially spaced. This provides us a useful tool toapproximate a signal by using its values at the node ( e kw ) . But practically it is difficult to have theexact sample value at the node ( e kw ) always. To overcome this problem, one can replace the value f ( e kw ) by the mean value of f ( e x ) in the interval (cid:2) kw , k + w (cid:3) , for k ∈ Z , w > . This idea motivatesus to define the Kantorovich-version of the generalized exponential sampling series (1.1) and thiswork is inspired from the classical Kantorovich version ([12, 13, 14, 23, 24, 25, 28, 37, 38, 35])of the generalized sampling series introduced by Butzer in [6] .The study of Kantorovich type generalizations of approximation operators is an importantsubject in approximation theory, as they can be used to approximate Lebesgue integrable func-tions. In the last few decades, Kantorovich modifications of several operators have been con-structed and their approximation behavior studied, we mention some of the work in this directione.g., [30, 2, 5, 3, 4, 1, 34, 32, 39, 31] etc.Let C ( R + ) be the space of all continuous and bounded functions on R + . A function f ∈ C ( R + ) is called 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 the space of alllog-uniformly continuous functions defined on R + by C ( R + ) . We consider M ( R + ) as the classof all Lebesgue measurable functions on R + and L ∞ ( R + ) as the space of all bounded functionson R + throughout this paper.For 1 ≤ p < + ∞ , let L p ( R + ) be the space of all the Lebesgue measurable and p -integrablefunctions defined on R + equipped with the usual norm k f k p . For c ∈ R , we define the space X c = { f : R + → C : f ( · )( · ) c − ∈ L ( R + ) } equipped with the norm k f k X c = k f ( · )( · ) c − k = Z + ∞ | f ( u ) | u c − du . The Mellin transform of a function f ∈ X c is defined byˆ M [ f ]( s ) : = Z + ∞ u s − f ( u ) du , ( s = c + it , t ∈ R ) . A function f ∈ X c ∩ C ( R + ) , c ∈ R is called Mellin band-limited in the interval [ − η , η ] , ifˆ M [ f ]( c + iw ) = | w | > η , η ∈ R + . et f : R + → C and c ∈ R . Then, Mellin differential operator θ c is defined by θ c f ( x ) : = x f ′ ( x ) + c f ( x ) , x ∈ R + . We consider θ f ( x ) : = θ f ( x ) throughout this paper. The Mellin differential operator of order r ∈ N is defined by θ c : = θ c , θ rc = θ c ( θ r − c ) . Let χ : R + → R be the kernel function which is continuous on R + such that it satisfies thefollowing conditions:(i) For every x ∈ R + , + ∞ ∑ k = − ∞ χ ( e − k x w ) = . (ii) M ( χ ) < + ∞ and lim γ → + ∞ ∑ | k − log ( u ) | > γ | χ ( e − k u ) | | k − log ( u ) | = , uniformly with respect to u ∈ R + . We define the algebraic moments of order ν for the kernel function χ as m ν ( χ , u ) : = + ∞ ∑ k = − ∞ χ ( e − k u )( k − log ( u )) ν , ∀ u ∈ R + . Similarly, the absolute moment of order ν can be defined as M ν ( χ , u ) : = + ∞ ∑ k = − ∞ | χ ( e − k u ) || k − log ( u ) | ν , ∀ u ∈ R + . We define M ν ( χ ) : = sup u ∈ R + M ν ( χ , u ) . Let x ∈ R + and w > . We define the Kantorovich version of the exponential sampling seriesgiven by (1.1) as follows: ( I χ w f )( x ) = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw f ( e u ) du , (1.2)where f : R + → R is locally integrable such that the above series is convergent for every x ∈ R + .It is clear that for f ∈ L ∞ ( R + ) , the above series is well defined for every x ∈ R + . The pa-per is organized as follows. In section 2 we prove the point-wise approximation theorem andVoronovskaya type asymptotic formula for the Kantorovich exponential sampling operators. Insection 3 we obtain the representation formula which gives the relationship between the expo-nential sampling operators with Kantorovich exponential sampling operators. The main resultof this section is inverse result for the operators (1.2). Finally we verify the assumptions used inthe theory by using Mellin B -splines and Mellin’s Fejer kernel and show the approximation offunctions by (1.2) graphically. Approximation Results
In this section, we obtain some direct results e.g. pointwise convergence theorem and Voronovskayatype asymptotic formula for the Kantorovich exponential sampling operators (1.2).
Theorem 2.1
Let f ∈ M ( R + ) ∩ L ∞ ( R + ) . Then, the series (1.2) converges to f ( x ) at every pointx ∈ R + , the point of continuity of f . Moreover, for f ∈ C ( R + ) lim w → ∞ k I χ w f − f k ∞ = . Proof.
Using the condition (i), we obtain | I χ w f ( x ) − f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw ( f ( e u ) − f ( x )) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∑ (cid:12)(cid:12) k − w log ( x ) (cid:12)(cid:12) < w δ + ∑ (cid:12)(cid:12) k − w log ( x ) (cid:12)(cid:12) ≥ w δ (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) w Z k + wkw | f ( e u ) − f ( x ) | du . : = I + I . Since f ∈ C ( R + ) , for every ε > δ > | f ( e u ) − f ( x ) | < ε , whenever | u − log ( x ) | < δ . Let w ′ be fixed in such a way that w < δ for every w > w ′ . Now, for u ∈ (cid:2) kw , k + w (cid:3) and w > w ′ and we have | u − log ( x ) | ≤ (cid:12)(cid:12)(cid:12) u − kw (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) kw − log ( x ) (cid:12)(cid:12)(cid:12) ≤ δ , whenever (cid:12)(cid:12) kw − log ( x ) (cid:12)(cid:12) < δ . This gives | I | < ε M ( χ ) . Similarly, we estimate I . | I | ≤ k f k ∞ ∑ (cid:12)(cid:12) k − w log ( x ) (cid:12)(cid:12) ≥ w δ (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) ≤ k f k ∞ ε . Combining the estimates of I − I , we get the desired result.Next, we derive the following asymptotic formula for the operators ( I χ w f ) w > . Theorem 2.2
Let f ∈ C ( ) ( R + ) and χ be the kernel function such that its first order momentvanishes for all u ∈ R + . Then, we have lim w → ∞ w (cid:2) ( I χ w f )( x ) − f ( x ) (cid:3) = ( θ f )( x ) . Proof.
For f ∈ C ( R + ) , the Taylor’s formula in terms of Mellin derivatives ([7]) upto secondorder term can be written as f ( e u ) = f ( x ) + ( θ f )( x )( u − log ( x )) + ( θ f )( x ) ( u − log ( x )) + h (cid:16) e u x (cid:17) ( u − log ( x )) , here h is a bounded function such that lim t → h ( t ) = . In view of (1.2) we obtain [( I χ w f )( x ) − f ( x )] = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw (cid:2) ( θ f )( x )( u − log ( x )) +( θ f )( x ) ( u − log ( x )) + h (cid:16) e u x (cid:17) ( u − log ( x )) i du : = I + I + I . First we evaluate I . I = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw (cid:2) ( θ f )( x )( u − log ( x )) (cid:3) du = ( θ f )( x ) + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw ( u − log ( x )) du = ( θ f )( x ) w . Now, we estimate I . I = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw h ( θ ( ) f )( x ) ( u − log ( x )) i du = ( θ ( ) f )( x ) + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw ( u − log ( x )) du = ( θ ( ) f )( x ) w (cid:0) + m ( χ , u ) + m ( χ , u ) (cid:1) . From the above estimates I and I , we obtain w ( I + I ) → ( θ f )( x ) as w → ∞ . Now I can bewritten as | I | = + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw (cid:2) h (cid:18) e u x (cid:19) ( u − log ( x )) (cid:3) du ≤ ∑ | k − w log ( x ) | < w δ (cid:12)(cid:12)(cid:12) χ ( e − k x w ) w Z k + wkw h h (cid:16) e u x (cid:17) ( u − log ( x )) i du (cid:12)(cid:12)(cid:12) + ∑ | k − w log ( x ) |≥ w δ (cid:12)(cid:12)(cid:12) χ ( e − k x w ) w Z k + wkw h h (cid:18) e u x (cid:19) ( u − log ( x )) i du (cid:12)(cid:12)(cid:12) : = I ′ + I ′′ . Since lim x → h ( x ) = , we have | wI ′ | ≤ ε w ( + M ( χ )) . Now using the fact that h ( x ) is bounded,we obtain | I ′′ | ≤ k h k ∞ ∑ | k − w log ( x ) |≥ w δ (cid:12)(cid:12) χ ( e − k x w ) w Z k + wkw ( u − log ( x )) du (cid:12)(cid:12) ≤ ( ε + ) w k h k ∞ . his gives | wI ′′ | ≤ ( ε + ) w k h k ∞ . Combining the estimates of I − I , we get the desired result. Corollary 2.1
The assumption that f is bounded on R + can be relaxed by assuming that thereare two positive constants α , β such that | f ( x ) | ≤ α + β | log ( x ) | , ∀ x ∈ R + . Proof.
First we show that the series (1.2) is well defined for such f . Indeed | I χ w f )( x ) | ≤ + ∞ ∑ k = − ∞ | χ ( e − k x w ) | w Z k + wkw | f ( e u | du ≤ + ∞ ∑ k = − ∞ | χ ( e − k x w ) | w Z k + wkw ( α + β | u | ) du ≤ ( α + β log x ) M ( χ ) + β + ∞ ∑ k = − ∞ | χ ( e − k x w ) | w Z k + wkw | u − log ( x ) | du ≤ (cid:18) α + β log ( x ) + β w (cid:19) M ( χ ) + β w M ( χ ) < + ∞ . This shows that the series ( I χ w f ) w > is absolutely convergent in R + . Now for any fixed a ∈ R + , we define P ( u ) : = f ( a ) + ( θ f )( a )( u − log ( a )) + ( θ ( ) f )( a ) ( u − log ( a )) . From the Taylor’sformula in terms of Mellin derivatives upto second order term, we can write as h (cid:18) e u a (cid:19) = f ( e u ) − P ( u )( u − log ( a )) , where h ( . ) is a bounded function such that lim t → h ( t ) = . This implies that h is bounded in theneighbourhood for | u − log ( a ) | < δ . For | u − log ( a ) | ≥ δ , we have | h ( e u a − ) | ≤ | f ( e u || u − log ( a ) | + | P ( u ) | u − log ( a ) | ≤ α + β | u || u − log ( a ) | + | P ( u ) | u − log ( a ) | . This shows that h ( . ) is bounded on R + . Now we can proceed in the similar manner as in Theo-rem 2.2, to get the same asymptotic formula.The logarithmic modulus of continuity is defined by ω ( f , δ ) : = sup {| f ( x ) − f ( y ) | : whenever | log ( x ) − log ( y ) | ≤ δ , δ ∈ R + } . The properties of logarithmic modulus of continuity can be seen in [16]. Now we obtain aquantitative estimate of the convergence of operator (1.2) for f ∈ C ( R + ) . heorem 2.3 Let f ∈ C ( R + ) . Then, we have | ( I χ w f )( x ) − f ( x ) | ≤ λ ω (cid:18) f , w (cid:19) , where λ = ( M ( χ ) + M ( χ )) . Proof.
We have | ( I χ w f )( x ) − f ( x ) | ≤ ∑ | k − w log ( x ) | < w δ | χ ( e − k x w ) | w Z k + wkw | f ( e u ) − f ( x ) | du + ∑ | k − w log ( x ) |≥ w δ (cid:12)(cid:12) χ ( e − k x w ) | w Z k + wkw | f ( e u ) − f ( x ) | du ≤ ∑ | k − w log ( x ) | < w δ (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12) ω ( f , | k − w log ( x ) | ) + k f k ∞ M ( χ ) ε ≤ ω ( f , δ ) (cid:18) M ( χ ) + w δ M ( χ ) (cid:19) . Choosing δ = w and λ = ( M ( χ ) + M ( χ )) , we get the desired estimate. In this section, we derive an inverse result of approximation for the Kantorovich exponentialsampling operators. In order to establish the saturation theorem, we first obtain a relation be-tween ( S χ w f ) w > and ( I χ w f ) w > for f ∈ C ( n ) ( R + ) . This work is motivated from Kantorovich typegeneralized sampling operators studied in [6, 26, 27, 11].
Theorem 3.1
Let f ∈ C ( n ) ( R + ) , n ∈ N . Then the following relation holds for every x ∈ R + : ( I χ w f )( x ) = n − ∑ j = ( S χ w θ ( j ) f ))( x )( j + ) ! + ˆ R wn ( x ) , (3.3) where ˆ R wn ( x ) : = n ! ∑ k ∈ Z χ ( e − k x w ) w (cid:20) Z k + / wk / w ( θ ( n ) f )( ξ )( u − k / w ) n du (cid:21) is absolutely convergentfor ξ ∈ ( e k / w , e k + / w ) and w > . Proof.
Using the n th order Mellin’s Taylor’s formula ([7]) and substituting x = e k / w , w > u ∈ (cid:2) k / w , ( k + ) / w (cid:3) , we have ξ ∈ ( e k / w , e k + / w ) . Thus weobtain f ( e u ) = f ( e k / w ) + ( θ f )( e k / w )( u − k / w ) + ( θ ( ) f )( e k / w ) ( u − k / w ) + ... + ( θ ( n − ) f )( ξ )( n − ) ! ( u − k / w ) n − + R wn , where R wn = ( θ ( n ) f )( ξ ) n ! ( u − log ( x )) n , ξ ∈ ( x , e u ) . ow we evaluate w Z k + / wk / w f ( e u ) du = w Z k + / wk / w (cid:2) f ( e k / w ) + ( θ f )( e k / w )( u − k / w ) + ... + R wn ( u ) (cid:3) = f ( e k / w ) + ( θ f )( e k / w ) w ( ) + ... + w Z k + / wk / w R wn ( u ) du . Using (1.2), we obtain ( I χ w f )( x ) = ( S χ w f )( x ) + ( ) w ( S χ w ( θ f ))( x ) + ... + ˆ R wn ( x ) = n − ∑ j = ( S χ w θ ( j ) f )( x )( j + ) ! w j + ˆ R wn ( x ) , where ˆ R wn ( x ) : = ∑ k ∈ Z χ ( e − k x w ) w Z k + / wk / w ( θ ( n ) f )( ξ ) n ! ( u − k / w ) n du . Next we show that the remain-der term ˆ R wn is absolutely convergent in R + . (cid:12)(cid:12) ˆ R wn (cid:12)(cid:12) ≤ n ! ∑ k ∈ Z (cid:12)(cid:12) χ ( e − k x w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w Z k + / wk / w ( θ ( n ) f )( ξ )( u − k / w ) n du (cid:12)(cid:12)(cid:12)(cid:12) ≤ k θ ( n ) f k ∞ ( n + ) ! 1 w n M ( χ ) < + ∞ . This completes the proof.Now, we have the following inverse result for the Kantorovich exponential sampling opera-tors ( I χ w f ) w > . Theorem 3.2
Let χ be the kernel function such that m ( χ , u ) = . Suppose that f ∈ C ( ) ( R + ) and k I χ w f − f k ∞ = o ( w − ) as w → ∞ . Then f is constant on R + . Proof.
Using the relation (3.3) for n = , we have | ( I χ w f )( x ) − f ( x ) | = | ( S χ w f )( x ) − f ( x ) + ˆ R w ( x ) | . Using the assumption that k I χ w f − f k ∞ = o ( w − ) , we obtain | ( S χ w f )( x ) − f ( x ) + ˆ R w ( x ) | = o ( w − ) , which implies that lim w → ∞ w [ S χ w f )( x ) − f ( x )] + lim w → ∞ w ˆ R w ( x ) = . (3.4)Consider | w ˆ R w ( x ) − ( θ f )( x ) | = | w ˆ R w ( x ) − ( S χ w θ f )( x ) + ( S χ w θ f )( x ) − ( θ f )( x ) |≤ | ( S χ w θ f )( x ) − ( θ f )( x ) | + | w ˆ R w ( x ) − ( θ f )( x ) | : = I + I . sing Theorem 5 and Corollary 2 in [19], we have | I | ≤ ε . Now we estimate I . | I | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) w + ∞ ∑ k = − ∞ χ ( e − k x w ) w Z k + wkw ( θ f )( ξ ) (cid:18) u − kw (cid:19) du (cid:21) − ( S χ w θ f )( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ w (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) + ∞ ∑ k = − ∞ χ ( e − k x w ) Z k + wkw (cid:18) u − kw (cid:19) h ( θ f )( ξ ) − ( θ f )( e kw ) i du (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . Since ξ ∈ (cid:2) kw , k + w ] and by using the continuity of ( θ f )( x ) , we have | ( θ f )( ξ ) − ( θ f )( e kw ) | < ε which gives | I | < ε M ( χ ) . Combining the estimates I − I , we obtainlim w → ∞ [ w ˆ R w ( x ) − ( θ f )( x )] = . Using the above estimate and (3.4), we have ( θ f )( x ) = ∀ x ∈ R + . This implies that f is constanton R + . In this section, we present few examples of the kernel functions based on the theory of Mellin’stransform which also satisfy the assumptions of the presented theory. The first example of kernelfunctions in this direction is the family of Mellin- B spline kernels [19].
The B-splines of order n in the Mellin setting for x ∈ R + are defined as¯ B n ( x ) : = ( n − ) ! n ∑ j = ( − ) j (cid:18) nj (cid:19)(cid:18) n + log ( x ) − j (cid:19) n + + . ¯ B n ( x ) is compactly supported for every n ∈ N . The Mellin transformation of ¯ B n (see [19]) isgiven by ˆ M [ ¯ B n ]( c + iw ) = (cid:18) sin ( w )( w ) ! n , w = . (4.5)To show that the above kernel satisfies the assumptions (i)-(ii), we use the Mellin’s - Poissonsummation formula [8]) which has the form ( i ) j + ∞ ∑ k = − ∞ χ ( e k x )( k − log ( u )) j = + ∞ ∑ k = − ∞ ˆ M ( j ) [ χ ]( k π i ) x − k π i , for k ∈ Z . The following lemma will be useful in this direction. emma 4.1 The condition + ∞ ∑ k = − ∞ χ ( e − k x w ) = is equivalent to the condition ˆ M [ χ ]( k π i ) = ( , if, k = , if, otherwise (4.6) Moreover, 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 . Proof.
The proof can be obtained by the similar arguments as given in the proof of Lemma 2 andLemma 3 in [6].Again from (4.5) we have,ˆ M [ ¯ B n ]( k π i ) = ( , if, k = , if, otherwise.This shows that ¯ B n ( x ) satisfies the condition (i). Since ¯ B n ( x ) is compactly supported, the condi-tion (ii) is also satisfied. By using Lemma 1 for j = , we obtain that the first order moment forˆ M [ ¯ B n ] vanishes in R + and hence it satisfies the assumption of Theorem 2.2. First we show theapproximation of f ( x ) by ( I χ w f ) w > , where f ( x ) = ( , ≤ x < − x , ≤ x < . Table 1
Error estimation (upto decimal points) in the approximation of f ( x ) by I χ w f ( x ) forw = , , . x | f ( x ) − I χ f ( x ) | | f ( x ) − I χ f ( x ) | | f ( x ) − I χ f ( x ) | . . . . . . . . . . . . . . . . .5 1 1.5 2 2.5 3 3.5−2−1.8−1.6−1.4−1.2−1−0.8−0.6−0.4−0.20 X−axis Y − a x i s I ( χ )I χ f (x) Figure 1: This figure exhibits the approximation of f ( x ) (Black) by the series ( I χ w f ) w > for w = ,
40 (Red and Green respectively) based on Mellin B-spline kernel.
The general form of Mellin’s-Fejer kernel is given by F c α ( x ) : = α π x c sinc (cid:18) απ log ( √ x (cid:19) , where c ∈ R , α > x ∈ R + . The Mellin’s transform for F c α is given byˆ M [ F c α ]( c + iw ) = ( | w | α , if, | w | ≤ α
0, if, | w | > α . The Mellin-Fejer kernel also satisfies the assumption (i)-(ii) analogously (see [19]), but it failsto satisfy the moment condition of Theorem 3, i.e, m ( χ , u ) = . This can be seen by using theequivalent condition mentioned in Lemma 4.1 for j = . Next we present the approximation ofthe function f ( x ) by the operator ( I χ w f ) w > , where f ( x ) = ( cos x , ≤ x < , ≤ x < . χ I χ I χ f (x) Figure 2: This figure exhibits the approximation of f ( x ) (Black) by the series ( I χ w f ) w > for w = , ,
80 (Green,Blue and Red respectively) based on Mellin-Fejer kernel F π ( x ) . Table 2
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