Convergence properties of spline-like cardinal interpolation operators acting on l p data
aa r X i v : . [ m a t h . F A ] D ec Convergence properties of spline-like cardinalinterpolation operators acting on l p data Jeff Ledford
Abstract If f ∈ { f ∈ L p ( R ) : f ( x ) = R π − π e ixξ dβ ( ξ ) , β ∈ B.V. ([ − π, π ]) } , then f isdetermined by its samples on the integers by taking an appropriate limit.Specifically, k f − L φ α f k L p ( R ) → α → ∞ provided that { φ α : α ∈ A } is what we call a spline-like family of cardinal interpolators. Keywords:
Cardinal Interpolation, Spline, Multiquadric, Gaussian,Poisson kernel
This paper continues the study of convergence properties of cardinal interpo-lation operators, which was popularized by the spline methods developed bySchoenberg and others. Ultimately, our goal is to prove a result that is simi-lar to the sampling theorem. This classical result shows that a Paley-Wienerfunction is determined by its values at the integers. Cardinal splines, so calledbecause they interpolate data on the integers, may be used to interpolate l data, then by letting the degree increase to infinity, the corresponding Paley-Wiener function is recovered pointwise. A similar result for a general class ofinterpolants was proved by the author in [4]. In [5], Marsden, Richards, andRiemenschneider showed that splines may also be used to recover functions fromthe space A = (cid:26) f ∈ L P ( R ) : f ( x ) = (2 π ) − / Z π − π e ixξ dβ ( ξ ) , β ∈ B.V. ([ − π, π ]) (cid:27) . (1)We wish to extend this result to a general class of interpolants, those that arebuilt from what we call spline-like cardinal interpolators.This remainder of this paper is organized into three sections, the first dealingwith definitions and preliminary facts concerning spline-like cardinal interpola-tors, the second proving the convergence result, and finally in the third sectionwe show that this class of interpolants includes odd-degree cardinal splines, theGaussian which was the subject of [6], Poisson kernels, and two families of mul-tiquadrics. Of these examples, the convergence property for the Poisson kernelsappears to be new, as does the result concerning the family of multiquadrics { ( x + c ) k − / : k ∈ N } . 1 Definitions and Basic Facts
We begin with a convention for the Fourier transform.
Definition 1.
The
Fourier transform of a function f ( x ) ∈ L ( R ) , is defined tobe the function ˆ f ( ξ ) = (2 π ) − / Z R f ( x ) e − ixξ dx. We make the usual extension to the class of tempered distributions usingthis convention. Our work will also make use of the mixed Hilbert transformwhich is described below, more details may be found in [5]. For x ∈ R , let m x be the integer such that m x − / ≤ x < m x + 1 /
2. If f = { f ( k ) } ∈ l p and1 < p < ∞ , we define the mixed Hilbert transform , denoted H [ f ]( x ), to be H [ f ]( x ) = X k = m x f ( k ) x − k . Proposition 1.3 in [5] provides a constant depending only on p such that k H [ f ] k L p ( R ) ≤ C p k f k l p (2)We need a definition similar to the one found in [4]. For our purposes, wemake the following definition. Definition 2.
We say that a function φ ( x ) satisfies the interpolating conditions if it satisfies the all of the following:(A1) φ ( x ) is a real valued slowly increasing function on R ,(A2) ˆ φ ( ξ ) ≥ and ˆ φ ( ξ ) ≥ m > in [ − π, π ] ,(A3) ˆ φ ( ξ ) ∈ C ( R \ { } ) , and(A4) there exists ǫ > such that for j = 0 , we have ˆ φ ( j ) ( ξ ) = O ( | ξ | − (1+ ǫ ) ) as | ξ | → ∞ . These conditions, assure us that the fundamental function L φ ( x ) defined byits Fourier transform via the formulaˆ L φ ( ξ ) = (2 π ) − / ˆ φ ( ξ ) X j ∈ Z ˆ φ ( ξ + 2 πj ) , (3)is continuous and solves the following interpolation problem. Problem 1.
Find a function L ∈ L ( R ) that satisfies L ( j ) = δ ,j for j ∈ Z . fundamental functions . By inter-polator , we mean the function φ from which a fundamental function is built.A fundamental function allows us to solve an l interpolation problem usinginterpolants of the form I φ [ f ]( x ) = X j ∈ Z f ( j ) L φ ( x − j ) , (4)where L φ ( x ) is defined in (3) and { f ( j ) } ∈ l . Henceforth, we focus on inter-polants of this form. Let us define an auxiliary function that will prove useful.Given a function f which satisfies (A1)-(A4) and k ∈ Z , we define M [ f ] k ( u ) = f ( u + 2 πk ) f ( u ) , | u | ≤ π. (5)Since we are interested in matters of convergence, we need to introduce aparameter. To this end we let A ⊂ (0 , ∞ ) be an unbounded index set whichcould be discrete or continuous depending on the example. We wish to makesure that our parameter interacts well with the limit. The following regularitydefinition makes use of the auxiliary function introduced in (5). Definition 3.
A collection of functions { φ α : α ∈ A } will be called a spline-likefamily of cardinal interpolators if the following conditions are satisfied:(B1) for all α ∈ A , φ α satisfies the interpolating conditions,(B2) X j ∈ Z X k = j k M [ ˆ φ α ] j ( ξ ) M [ ˆ φ ′ α ] k ( ξ ) k L ([ − π,π ]) ≤ C , independent of α ,(B3) for j = 0 , lim α →∞ M [ ˆ φ α ] j ( ξ ) = 0 for almost every | ξ | ≤ π , and(B4) there exists { M j } ∈ l such that M [ ˆ φ α ] j ( ξ ) ≤ M j for j = 0 , α ∈ A . This definition seems quite restrictive, nevertheless many popular choices ofinterpolator satisfy these conditions. The reader interested in examples mayskip ahead to the final section.We turn now to preliminary results which will aid in the proof of the maintheorem. Most of these are straightforward consequences of the conditions listedabove. For the remainder of the paper, we fix a particular spline-like family ofcardinal interpolators { φ α : α ∈ A } , and note that the particular value of theconstant C depends on its occurrence and may change from one line to the next. Lemma 1. If f = { f ( k ) } ∈ l , L φ α ( x ) is defined by (3) , and I φ α [ f ]( x ) is definedby (4) , then for each α ∈ A , L φ α ∈ L ( R ) ∩ C ( R ) and I φ α [ f ] ∈ L ( R ) ∩ C ( R ) .Proof. To see that L φ α ∈ L ( R ) ∩ C ( R ), we need only consider the Fouriertransform. Z R | ˆ L φ α ( ξ ) | dξ = X j ∈ Z Z π − π | ˆ L φ α ( ξ − πj ) | dξ Z π − π X j ∈ Z | ˆ L φ α ( ξ − πj ) | dξ = (2 π ) / Thus L φ α ∈ C ( R ). Since φ α satisfies (A2), the calculation for L φ α ∈ L ( R ) issimilar. To see that I φ α [ f ] ∈ L ( R ), we again use the Fourier transform. Z R | \ I φ α [ f ]( ξ ) | dξ = Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ L φ α ( ξ ) X k ∈ Z f ( k ) e ikξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = X j ∈ Z Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ L φ α ( ξ − πj ) X k ∈ Z f ( k ) e ikξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≤ Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ Z f ( k ) e ikξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = k{ f ( k ) }k l The last equality coming from Parseval’s formula. A similar argument showsthat \ I φ α [ f ] ∈ L ( R ), hence I φ α [ f ] ∈ C ( R ). Lemma 2.
For all α ∈ A , L φ α ( x ) satisfies L φ α ( j ) = δ ,j for j ∈ Z .Proof. Using the inversion formula for the Fourier transform, we get for j ∈ Z : L φ α ( j ) = (2 π ) − / Z R ˆ L φ α ( ξ ) e ijξ dξ = (2 π ) − X j ∈ Z Z π − π ˆ L φ α ( ξ − πj ) e ijξ dξ = (2 π ) − Z π − π e ijξ dξ = δ ,j . We introduce an operator which plays an important role throughout theremainder of the paper, the Whittaker map W : l p → L p ( R ), given by W [ { c k } ]( x ) = X k ∈ Z c k sin( π ( x − k )) π ( x − k ) . We mention a few results concerning W . In [5], it is shown (in Lemma 1.4),that this a continuous mapping for 1 < p < ∞ , i.e. k W [ { c k } ]( x ) k L p ( R ) ≤ C p k{ c k }k l p , (6)where C p depends only on p . Furthermore, it is shown (in Theorem 3.4) thatthe space B = { f : f ( x ) = W [ { c k } ]( x ) , { c k } ∈ l p } is equivalent to the space A defined in (1).4 Main Result
Our goal is to prove a generalization of Theorem 3.4 in [5], which gives anotherequivalent formulation of the space A . Our argument closely resembles the onegiven there. We begin by finding the L p -norm of L φ α . Lemma 3. If { φ α : α ∈ A } is a spline-like family of cardinal interpolants, thenthere exists a constant C independent of α such that k D ˆ L φ α k L ( R ) ≤ C (7) Proof.
The quotient rule gives us D ˆ L φ α ( ξ ) = (2 π ) − / ˆ φ ′ α ( ξ ) P α ( ξ ) − ˆ φ α ( ξ ) P ′ ( ξ ) P α ( ξ ) , where P α ( ξ ) = P j ∈ Z ˆ φ α ( ξ − πj ). We have(2 π ) / Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ φ ′ α ( ξ ) P α ( ξ ) − ˆ φ α ( ξ ) P ′ ( ξ ) P α ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = X m ∈ Z Z (2 m +1) π (2 m − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ φ ′ α ( ξ ) P α ( ξ ) − ˆ φ α ( ξ ) P ′ ( ξ ) P α ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = X m ∈ Z Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ φ ′ α ( ξ + 2 πm ) P α ( ξ ) − ˆ φ α ( ξ + 2 πm ) P ′ ( ξ ) P α ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ. The integrand may be simplified by letting u α ( ξ ) = P j =0 ˆ φ α ( ξ − πj ) for | ξ | ≤ π .Then (A3) and (A4) imply that we can differentiate term by term, so we get X m ∈ Z Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ φ ′ α ( ξ + 2 πm ) P α ( ξ ) − ˆ φ α ( ξ + 2 πm ) P ′ ( ξ ) P α ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = X m ∈ Z Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ φ ′ α ( ξ + 2 πm ) u α ( ξ ) − ˆ φ α ( ξ + 2 πm ) u ′ α ( ξ ) P α ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≤ Z π − π X m ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ φ ′ α ( ξ + 2 πm ) u α ( ξ + 2 πm ) − ˆ φ α ( ξ + 2 πm ) u ′ α ( ξ + 2 πm )ˆ φ α ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≤ Z π − π [ ˆ φ α ( ξ )] − X m ∈ Z | ˆ φ α ( ξ + 2 πm ) X j =0 ˆ φ ′ α ( ξ + 2 π ( m − j )) | dξ ≤ Z π − π X m ∈ Z X n = m | M [ ˆ φ α ] m ( ξ ) M [ ˆ φ ′ α ] n ( ξ ) | dξ =2 X m ∈ Z X n = m Z π − π | M [ ˆ φ α ] m ( ξ ) M [ ˆ φ ′ α ] n ( ξ ) | dξ ≤ C The constant is independent of α from (B2). The interchange of the order ofthe sums and the integral is justified by Tonelli’s theorem.5e have the following two straightforward corollaries. Corollary 1.
If the conditions of the above lemma are met, then we have thepointwise bound | L φ α ( x ) | ≤ C (1 + | x | ) − , (8) where C is independent of α .Proof. This follows from taking the Fourier transform of (1 + x ) L φ α ( x ). Corollary 2.
For < p < ∞ , k L φ α k L p ( R ) ≤ C p , where C p is independent of α .Proof. This follows immediately from the above corollary.We may now establish a bound for k I φ α k l p → L p . Proposition 1.
Let { φ α : α ∈ A } be a spline-like family of cardinal interpola-tors, f = { f ( k ) } ∈ l p , and < p < ∞ , then k I φ α [ f ] k L p ( R ) ≤ C p k f k l p , (9) where I φ α is defined as in (4) and C p is independent of both f and α .Proof. The method of proof is similar to Theorem 3.1 in [5], we encouragethe reader to consult that proof as well. We will use the converse of H¨older’sinequality. To this end, we suppose that 1 < p < ∞ and f = { f ( k ) } ∈ l p satisfies f ( k ) = 0 for | k | > N . Letting q be the conjugate exponent to p , i.e.1 /p + 1 /q = 1, we suppose that g ∈ L q ( R ) is supported in [ − R, R ]. Recallingthat m x is the integer which satisfies m x − / ≤ x < m x + 1 /
2, we have (cid:12)(cid:12)(cid:12)(cid:12)Z R I φ α ( x ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R − R X | j |≤ N f ( j ) L φ α ( x − j ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R − R | f ( m x ) | L φ α ( x − m x ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R − R X j = m x f ( j ) L φ α ( x − j ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C p k f k l p k g k L q ( R ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R − R X j = m x f ( j ) L φ α ( x − j ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Here we have used Corollary 2 along with H¨older’s inequality to estimate thefirst term, we note that the constant C p is independent of α . It remains tobound the second term. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R − R X j = m x f ( j ) L φ α ( x − j ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π ) − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R − R X j = m x f ( j ) Z R ˆ L φ α ( ξ ) e i ( x − j ) ξ dξg ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =(2 π ) − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R − R X j = m x f ( j ) x − j Z R e ixξ D ˆ L φ α ( ξ ) dξg ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z R − R | H [ f ]( x ) g ( x ) | dx ≤ C ∗ p k f k l p k g k L q ( R ) Here we have used the Fourier representation of L φ α and integrated by parts,then used the bound (7) along with H¨older’s inequality and (2). The constant C ∗ p is independent of α , so combining the two estimates and taking N, R → ∞ completes the proof.
Proposition 2.
Let { φ α : α ∈ A } be a spline-like family of cardinal interpola-tors, f = { f ( j ) } ∈ l p , and < p < ∞ , then we have lim α →∞ k I φ α [ f ]( x ) − W [ f ]( x ) k L p ( R ) = 0 . (10) Proof.
From Corollary 2, we may apply the uniform boundedness theorem. Thusit is enough to check the result on y j = { δ k,j : k ∈ Z } . We have k I φ α [ y j ]( x ) − W [ y j ]( x ) k L p ( R ) = k L φ α ( x − j ) − sin( π ( x − j )) π ( x − j ) k L p ( R ) = k L φ α ( x ) − sin( πx ) πx k L p ( R ) Corollary 3 combined with the dominated convergence theorem will finish theproof, provided that we show the pointwise convergence. We will use the Fouriertransform. We have | L φ α ( x ) − sin( πx ) πx | = (2 π ) − / (cid:12)(cid:12)(cid:12)(cid:12)Z R ˆ L φ α ( ξ ) e ixξ dξ − Z π − π e ixξ dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 π ) − / Z π − π | ˆ L φ α ( ξ ) − | dξ + X j =0 Z π − π | ˆ L φ α ( ξ − πj ) | dξ . We will estimate the terms in parentheses separately, for the first term we have Z π − π | ˆ L φ α ( ξ ) − | dξ = Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u α ( ξ )ˆ φ α ( ξ ) + u α ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≤ Z π − π X j =0 M [ ˆ φ α ] j ( ξ ) dξ. The function u α ( ξ ) is the same one introduced in the proof of Lemma 3. Since(B3) and (B4) hold, the dominated convergence theorem shows that this term7ends to 0. The calculation for the second term is similar: X j =0 Z π − π | ˆ L φ α ( ξ − πj ) | dξ ≤ Z π − π X j =0 M [ ˆ φ α ] j ( ξ ) dξ. Again, the terms tend to 0 by the dominated convergence theorem. Hence wehave shown that lim α →∞ (cid:12)(cid:12)(cid:12)(cid:12) L φ α ( x ) − sin( πx ) πx (cid:12)(cid:12)(cid:12)(cid:12) = 0 , which completes the proof.These propositions lead us to our main result, stated in the theorem below. Theorem 1.
Let < p < ∞ and { φ α : α ∈ A } be a spline-like family ofcardinal interpolators, then the following spaces are equivalent. A = n f ∈ L p ( R ) : f ( x ) = R π − π e ixξ dβ ( ξ ) , β ∈ B.V. ([ − π, π ]) o B = { f : f ( x ) = W [ { c k } ]( x ) , { c k } ∈ l p }C = n f : f ( x ) = L p lim α →∞ I φ α [ { c k } ]( x ) , { c ( k ) } ∈ l p o Proof.
That A and B are equivalent is shown in Theorem 3.4 in [5]. The previousproposition shows that B and C are equivalent.We have the following corollary concerning the recovery of functions fromthe space A . Corollary 3. If f ∈ A and { φ α : α ∈ A } is a spline-like family of cardinalinterpolators, then lim α →∞ | f ( x ) − I φ α [ { f ( k ) } ]( x ) | = 0 , and the convergence is uniform.Proof. Theorem 1 implies that f ( x ) = W [ { f ( k ) } ]( x ), hence the pointwise esti-mate from Proposition 2 provides the limit, as well as uniformity. In this section we provide examples of spline-like families of cardinal interpola-tors. Some of these results appear in the literature. Specifically, the spline ana-log of the L p result that is obtained here was worked out by Marsden, Richards,and Riemenschneider in [5] and the Gaussian analog was worked out by Riemen-schneider and Sivakumar in [6]. The work done in [4] for the space L suggestedthat an appropriate generalization to L p should exist.The interpolating conditions (A1)-(A4), are generally straightforward exer-cises which we leave to the reader. Most of the calculations involved are routine,8o we show only a typical calculation. Checking (B2) is the most tedious cal-culation and to ease out work we split the double sum into five pieces definedbelow.Λ = { (0 , n ) ∈ Z : n = 0 } Λ = { ( m, n ) ∈ Z : m > , n > , n = m } Λ = { ( m, n ) ∈ Z : m < , n > } Λ = { ( m, n ) ∈ Z : m < , n < , n = m } Λ = { ( m, n ) ∈ Z : m > , n < } These correspond more or less to the quadrants so that we can break up absolutevalues easily. Finally, we will omit constants from the various transforms toavoid clutter and they cancel out in the fundamental function in any case.
We will examine the spline interpolator whose definition may be found in [5].Owing to the positivity requirement in (A2), we will only consider odd degreesplines,i.e. those which reduce to an odd degree polynomial off of the integers,whose Fourier transforms are given by ˆ φ k ( ξ ) = | ξ | − k for k ∈ N . The reader isencouraged to examine [5] for more details. We begin by verifying (B2) on Λ : X (0 ,n ) ∈ Λ Z π − π | M [ ˆ φ k ] ( ξ ) M [ ˆ φ ′ k ] n ( ξ ) | dξ ≤ X n =0 Z π − π | ξ | k (2 k ) | ξ + 2 πn | − k − dξ ≤ X n =0 (2 | n | − − k ≤ X n =0 (2 | n | − − . This sum is independent of k ∈ N , so we move on to Λ . X ( m,n ) ∈ Λ Z π − π | M [ ˆ φ k ] m ( ξ ) M [ ˆ φ ′ k ] n ( ξ ) | dξ ≤ X m> X n> (2 | m | − − k (2 | n | − − k ≤ X m> (2 | m | − − ! Analogous calculations show (B2) is satisfied for the remaining sums. To check(B3), we suppose j = 0 and | ξ | < π , then we have π < | πj + ξ | so that M [ ˆ φ k ] j ( ξ ) ≤ (2 j + ξ/π ) − k j = 0, we have M [ ˆ φ k ] j ( ξ ) ≤ (2 | j | − − . Before we move to our next example a word on necessity is in order. Ourconditions (B1)-(B4) cannot be necessary since even order splines do not meetthe requirements, nevertheless, convergence is possible as shown in [5]. Thecondition that is satisfied is that the sum in the denominator of ˆ L is boundedaway from 0, which is the purpose of (A2). We now consider { exp( − x / (4 α )) : α ≥ } , which is the focus of [6], although wehave changed the parameter to suit our purposes. We have ˆ φ α ( ξ ) = exp( − αξ ),so that (B1) is clear. As for (B2), we again only show Λ and Λ since the othercases are similar. X (0 ,n ) ∈ Λ Z π − π | M [ ˆ φ α ] ( ξ ) M [ ˆ φ ′ α ] n ( ξ ) | dξ ≤ X n =0 Z π − π α exp( απ ) | ξ + 2 πn | exp( − α ( ξ + 2 πn ) ) dξ ≤ X n =0 exp( − απ ( | n | − | n | )) ≤ X n =0 exp( − π ( | n | − | n | )) X ( m,n ) ∈ Λ Z π − π | M [ ˆ φ α ] m ( ξ ) M [ ˆ φ ′ α ] n ( ξ ) | dξ ≤ X m,n> exp( − απ ( m − m )) exp( − απ ( n − n )) ≤ X m> exp( − π ( m − m )) ! Conditions (B3) and (B4) follow from similar reasoning as used for odd-degreesplines.
We now consider { ( x + α ) − : α ≥ } , the family of Poisson kernels, whoseFourier transforms are given by ˆ φ α ( ξ ) = exp( − α | ξ | ). Since (B1) is clear, wecheck (B2) for Λ and Λ . X (0 ,n ) ∈ Λ Z π − π | M [ ˆ φ α ] ( ξ ) M [ ˆ φ ′ α ] n ( ξ ) | dξ X n =0 Z π − π α exp( απ ) exp( − α | ξ + 2 πn | ) dξ ≤ X n =0 exp( − πα ( | n | − ≤ X n> exp( − πn ) X ( m,n ) ∈ Λ Z π − π | M [ ˆ φ α ] m ( ξ ) M [ ˆ φ ′ α ] n ( ξ ) | dξ ≤ X m,n> Z π − π α exp(2 απ ) exp − α ( ξ + 2 πm ) exp( − α ( ξ πn )) dξ ≤ X m,n> exp( − απ ( m − − απ ( n − ≤ X m> exp( − πm ) ! To check (B3) we note that for | ξ | < π and j = 0, π < | πj + ξ | so we have M [ ˆ φ α ] j ( ξ ) ≤ exp( α ( π − | πj + ξ | )) . Since the exponent is negative, the right hand side tends to 0 as required. For(B4) we have M [ ˆ φ α ] j ( ξ ) ≤ exp( − π ( | j | − . We consider the family of multiquadrics { ( x + c ) k − / : k ∈ N } for a fixed c >
0. These kernels correspond to ‘smoothed out’ odd degree splines and theconvergence result for these interpolation operators appears to be new. We needthe Fourier transforms which may be found in [3]; neglecting constants we haveˆ φ k ( ξ ) = | ξ | − k K k ( c | ξ | ), where K a ( u ) = Z ∞ exp( − u cosh( t )) cosh( at ) dt, u > Lemma 4.
For | a | ≥ / and u > , we have the inequalities ( π/ / u − / exp( − u ) ≤ K a ( u ) ≤ (2 π ) / u − / exp( − u ) exp( a / (2 u )) . (11) Proof.
This is just a combination of Corollary 5.12 and Lemma 5.13 in [7].These bounds also appeared in [1] as asymptotic relations.11he interpolatory conditions follow easily from the definition of the Macdon-ald function, also of interest is that ˆ φ k ( ξ ) decreases on (0 , ∞ ). It is worthwhileto start with (B4), as we will see that the cases k = 1 and k > k = 1 and use (11). M [ ˆ φ ] j ( ξ ) ≤ / (2 π )) exp( c ( π − | ξ + 2 πj | )) ( π/ | ξ + 2 πj | ) / ≤ | j | − − / For k > M [ ˆ φ k ] j ( ξ ) ≤ (2 | j | − − k Thus we see that (B4) is satisfied with M j = 3(2 | j |− − / . The above estimatesimply that (B3) holds as well. We now must check (B2). Again we split up ourestimate into k = 1 and k >
1. For k = 1, our estimate of the Λ sum is similarto the calculation for (B2), indeed we have X (0 ,n ) ∈ Λ Z π − π | M [ ˆ φ ] ( ξ ) M [ ˆ φ ′ ] n ( ξ ) | dξ ≤ X n =0 / (2 π )) exp( − cπ ( | n | − | n | − − / ≤ X n =0 (2 | n | − − / As for k >
1, we use the corresponding estimate and see that X (0 ,n ) ∈ Λ Z π − π | M [ ˆ φ k ] ( ξ ) M [ ˆ φ ′ k ] n ( ξ ) | dξ ≤ X n =0 (2 | n | − − k ≤ X n =0 (2 | n | − − So we can see that the sum over Λ is bounded independent of k . For the sumover Λ and k = 1, we have X ( m,n ) ∈ Λ Z π − π | M [ ˆ φ ] m ( ξ ) M [ ˆ φ ′ ] n ( ξ ) | dξ ≤ X m> (2 m − − / ! , and for k > X ( m,n ) ∈ Λ Z π − π | M [ ˆ φ k ] m ( ξ ) M [ ˆ φ ′ k ] n ( ξ ) | dξ X m> (2 m − − k ! ≤ X m> (2 m − − ! . Hence the sum of Λ is also bounded independent of k . Analogous bounds holdfor the other sums.If { a k : k ∈ N } ⊆ [1 / , ∞ ) \ N satisfies dist( { a k } , N ) >
0, then a slightmodification of the argument given above will show that for c > { ( x + c ) a k : k ∈ N } is a spline-like family of cardinal interpolators as well. We will now consider the family of multiquadrics { ( x + c ) a : c ≥ } , where a ∈ R is fixed. The case that a = 1 / φ c ( ξ ) = | ξ | − ( a +1 / K a +1 / ( c | ξ | )where a / ∈ ˜ N = N ∪ { } ∪ {− k − / k ∈ N } . In order to use (11), we continuewith a ∈ ( R \ ˜ N ) ∩ { a ∈ R : | a + 1 / | ≥ / } . As usual, we check (B2), (B3),and (B4). We check (B4) first. M [ ˆ φ c ] j ( ξ ) ≤ / (2 π )) exp( c ( π − | ξ + 2 πj | )) ( π/ | ξ + 2 πj | ) a +1 / ≤ | j | + 1) | a | +1 / exp( − cπ ( | j | − ≤ | j | + 1) | a | +1 / exp( − π ( | j | − | ξ | < π and j = 0, we can see that π − | ξ + 2 πj | <
0, hence the exponentin the above calculation is negative and all of the terms tend to 0 as c → ∞ ,which means (B3) is satisfied. We must now check (B2). For the sum over Λ ,we have X (0 ,n ) ∈ Λ Z π − π | M [ ˆ φ c ] ( ξ ) M [ ˆ φ ′ c ] n ( ξ ) | dξ ≤ a / (2 π )) X n =0 (2 | n | + 1) a +1 / exp( − cπ ( | n | − ≤ a / (2 π )) X n =0 (2 | n | + 1) a +1 / exp( − π ( | n | − , which is bounded uniformly in c . For Λ , we have X ( m,n ) ∈ Λ Z π − π | M [ ˆ φ c ] m ( ξ ) M [ ˆ φ ′ c ] n ( ξ ) | dξ ≤ a /π ) X m> (2 j + 1) a +1 / exp( − cπ ( m − ! ≤ a /π ) X m> (2 j + 1) a +1 / exp( − π ( m − ! Since analogous estimates hold for the other sums, this collection of multi-quadrics is also a spline-like family of interpolators.13 eferences [1] M. Abramowitz I. Stegun, eds.,