Approximation of a function and its derivatives by entire functions
aa r X i v : . [ m a t h . C V ] A ug APPROXIMATION OF A FUNCTION AND ITSDERIVATIVES BY ENTIRE FUNCTIONS
P. M. GAUTHIER AND J. KIENZLE
Abstract.
A simple proof is given for the fact that, for m anon-negative integer, a function f ∈ C ( m ) ( R ) , and an arbitrarypositive continuous function ǫ, there is an entire function g, suchthat | g ( i ) ( x ) − f ( i ) ( x ) | < ǫ ( x ) , for all x ∈ R and for each i =0 , · · · , m. We also consider the situation, where R is replaced byan open interval. Introduction
For an open interval I = ( a, b ) , −∞ ≤ a < b ≤ + ∞ , and m =0 , , · · · , denote by C ( m ) ( I ) the space of functions f : I → C , whose de-rivitaves f (0) , f (1) , · · · , f ( m ) exist and are continuous on I. For a closedinterval I, let C ( m ) ( I ) be the space of functions f ∈ C ( m ) ( I ) , suchthat f (0) , f (1) , · · · , f ( m ) extend continuously to I. By abuse of nota-tion, we continue to denote these extensions by f ( j ) respectively. Thefollowing generlalization of the Weierstrass approximation theorem iswell-known. Theorem 1.
For −∞ < a < b < + ∞ , and m a non-negative integer,let f ∈ C ( m ) ([ a, b ]) and ǫ > . Then, there is a polynomial p, such that | p ( i ) ( x ) − f ( i ) ( x ) | < ǫ, for all x ∈ [ a, b ] and i = 0 , , · · · , m. To prove this theorem, we merely approximate f ( m ) by a polynomialand integrate m times.An other extension of the Weierstrass theorem, not as well knownas it should be, is the following theorem of Carleman [1], in which abounded interval is replaced by the entire real line. Denote by C + ( X )the positive continuous functions on a set X. Theorem 2. (Carleman) Let f ∈ C ( R ) and ǫ ∈ C + ( R ) . Then, thereexixts an entire function g such that | f ( x ) − g ( x ) | < ǫ ( x ) , x ∈ R . Mathematics Subject Classification.
Primary: 30E10.
Key words and phrases.
Carleman theorem.Research of second author supported by NSERC (Canada).
Note that f can be approximated much better than uniformly, since ǫ ( x ) may decrease to zero with arbitrary speed, as x → ∞ . Of course,since every continuous function on a bounded closed interval extendscontinuously to R and since entire functions are represented by theirMaclaurin series, the Weierstrass theorem is contained in the Carle-man theorem. There are many proofs of the Weierstrass theorem invarious textbooks on approximation, but the original proof of Weier-strass actually used a preliminary version of the Carleman theorem.That is, Weierstrass began by approximating a continuous function ona bounded closed interval by entire functions and then approximatingthe entire function by partial sums of its Maclaurin series.Let I = ( a, b ) be an interval, −∞ ≤ a ≤ b ≤ + ∞ . Denote by I c = R \ I the complement of I in R . For an open subset U ⊂ C , wedenote by H ( U ) the family of functions holomorphic on U . Theorem 3.
Let m be a non-negative integer, I = ( a, b ) , f ∈ C m ( I ) , and ǫ ∈ C + ( I ) . Then, there exists a function g ∈ H ( C \ I c ) , such that | f ( i ) ( x ) − g ( i ) ( x ) | < ǫ ( x ) , x ∈ I, i = 0 , , · · · , m. In particular, we have the following generalization of Theorem 1, dueto Hoischen [6, Satz 3].
Corollary 4.
Let m be a non-negative integer, f ∈ C m ( R ) , and ǫ ∈ C + ( R ) . Then, there exists an entire function g, such that | f ( i ) ( x ) − g ( i ) ( x ) | < ǫ ( x ) , x ∈ R , i = 0 , · · · , m. The results of Carleman and Hoischen have been extended in variousdirections. For example, Carleman’s theorem was extended by Schein-berg to approximation by entire functions of several complex variablesand Frih and Gauthier [3, Corollary] showed the corresponding exten-sion of the theorem of Hoischen on the simultaneous approximation ofderivatives. In [9] and [3] the functions to be approximated are definedon the real part R N of C N = R N + i R N and they are approximated byfunctions holomorphic in all of C N . Very recently, Johanis [7] has con-sidered the more general problem of approximating a function f givenon only a portion Ω of R N . Whitney’s famous theorem [10] allows oneto approximate such functions f by functions analytic on Ω . Of courseevery function analytic on Ω naturally extends holomorphically to aneighborhood of Ω in C N , but this neighborhood will depend on theanalytic function. The beautiful result of Johannis shows that there isa domain e Ω ⊂ C N , depending only on Ω and not on f, such that f canbe approximated by functions holomorphic on e Ω . When applied to oursituation, where N = 1 and Ω is an interval I, the domain e I is smallerthan the domain C \ I c which we obtain in Theorem 3. PPROXIMATION BY ENTIRE FUNCTIONS 3
For a closed set E ⊂ C , let A ( E ) ≡ C ( E ) ∩ H ( E o ). In the Carlemantheorem, if we replace the real line R by a a closed subset E ⊂ C , then the function f to be approximated must be, not only continuouson E, but also holomorphic on the interior of E. That is, f must liein A ( E ) . A condition on sets E, necessary for the possibility of suchapproximations, was introduced in [5], and in [8] this condition wasshown to be also sufficient.The techniques employed in previous papers are quite technical. Theaim of the present note is to show that Theorem 3, extending Theorem1 to open intervals and ǫ decreasing to zero with arbitrary speed, canbe proved in the same way as the elementary proof of Theorem 1, thatis, by approximating the derivative of highest order and integrating.2. Preliminaries
A fundamental lemma, known as the Walsh Lemma, asserts that, fora compact set K ⊂ C , if every function f ∈ A ( K ) can be uniformlyapproximated by rational functions having no poles on K, then, notonly are there rational functions which uniformly approximate f, thereare even rational functions which, in addition to approximating f, alsosimultaneously interpolate f at finitely many given points of K. ThisWalsh Lemma has been extended to the context of functional analysis.For a topological vector space X , we denote by X ∗ the (continuous)dual. If X and Y are topological vector spaces, where X is a subspaceof Y , then of course Y ∗ ⊂ X ∗ . The following result on simultaneous ap-proximation and interpolation is a generalization of the Walsh Lemmadue to Deutsch [2]. Lemma 5.
Let X be a dense subspace of a normed vector space Y . Let y ∈ Y , ǫ be a positive number and L , · · · , L n ∈ Y ∗ . Then, there exists x ∈ X such that | y − x | < ǫ and L i ( y ) = L i ( x ) , i = 1 , · · · , n. A compact set K ⊂ C is said to be a set of polynomial approxima-tion, if for each f ∈ A ( K ) and ǫ > , there exists a polynomial p suchthat | f − p | < ǫ . The celebrated theorem of Mergelyan (see [4]) statesthat a compact set is a set of polynomial approximation if and only ifits complement is connected. A particular case of the Walsh lemma isthe following. Lemma 6.
Let K ⊂ C be a compact set of polynomial approximation.Then, for all φ ∈ A ( K ) , L i ∈ A ( K ) ∗ , i = 1 , · · · , n and for all ǫ > , there exists a polynomial p, such that | φ − p | < ǫ and L i ( φ ) = L i ( ψ ) , i =1 , · · · , n. P. M. GAUTHIER AND J. KIENZLE
For φ ∈ C ([ j, j − , j ∈ Z , set T j ( φ ) = R jj − R x R x · · · R x m − φ ( t ) dtdx m − · · · dx ,T j ( φ ) = R jj − R x R x · · · R x m − φ ( t ) dtdx m − · · · dx ,...T mj ( φ ) = R jj − φ ( t ) dt. Lemma 7.
For all f ∈ C ( R ) and for all ǫ ∈ C + ( R ) , there existsan entire function g such that g (0) = f (0); T ij ( g ) = T ij ( f ) for i =1 , , · · · , m and j ∈ Z ; and | f ( t ) − g ( t ) | < ǫ ( t ) .Proof. First of all, we can see that Lemma 7 is true for a finite numberof j, by applying Lemma 6 to a closed interval E containing the intervals[ j − , j ] in question.Let f ∈ C ( R ) and ǫ ∈ C + ( R ) . We may assume that ǫ ( t ) = ǫ ( | t | )and ǫ ( | t | ) is decreasing as | t | grows. Let { ǫ k } be a sequence of positivenumbers such that ǫ k < ǫ ( k ) and P ∞ k = ℓ ǫ k < ǫ ( t ) / ℓ ≥ | t | , t ∈ R .We may choose, ǫ k = ǫ ( k ) / k +2 . Indeed, ∞ X k = ℓ ǫ k = ∞ X k = ℓ ǫ ( k ) / k +2 ≤ ǫ ( ℓ ) ∞ X k = ℓ / k +2 = ǫ ( ℓ ) / ℓ +1 ≤ ǫ ( t ) / . Now, for each k ∈ N , set E k = D k − S [ − k, − ( k − S [ k − , k ] , where D r is the disc of center 0 and radius r. By Lemma 6 and the Weierstrass approximation theorem, there ex-ists a polynomial g such that | f − g | < ǫ on [ − , , f ( j ) = g ( j ) , for j = − , , T ji ( f ) = T ji ( g ) , i = 1 , · · · , m ; j = 0 , . Set h = ( g on D f on [ − , \ [ − , . Since h ∈ A ( E ) , it follows from Lemma 6 and the Mergelyan theoremthat there is a polynomial g such that | h − g | < ǫ on E , h ( j ) = g ( j ) for j = − , − , · · · , T ji ( h ) = T ji ( g ) , i = 1 , · · · , m ; j = − , , · · · , . Thus, we have T ji ( f ) = T ji ( g ) , i = 1 , · · · , m ; j = − , , · · · , f ( j ) = g ( j ) for j = − , − , · · · , PPROXIMATION BY ENTIRE FUNCTIONS 5 and | f − g | < ( ǫ on [ − , \ [ − , ǫ + ǫ on [ − , . Indeed, | f − g | ≤ | f − h | + | h − g | = ( | h − g | < ǫ on [ − , \ [ − , | f − g | + | h − g | < ǫ + ǫ on [ − , . We also have that | g − g | < ǫ on D . Indeed, | g − g | ≤ | g − h | + | h − g | < ǫ + 0 on D . Setting g o = g , we shall show by induction that for k = 1 , , · · · , there exist polynomials g k , such that(1) T ji ( f ) = T ji ( g k ) , i = 1 , · · · , m ; j = − ( k − , · · · , k. (2) f ( j ) = g k ( j ) for j = − k, · · · , k (3) | f − g k | < ǫ k on [ − k, k ] \ [ − ( k − , k − ǫ k − + ǫ k on [ − ( k − , k − \ [ − ( k − , k − ...ǫ + ǫ + · · · + ǫ k on [ − , | g k − g k − | < ǫ k on D k − . As shown before, we already verified the cases k = 1 and 2. Wesuppose the validity of the cases k = 1 , · · · , n . Set h n +1 = ( g n on D n f on [ − ( n + 1) , n + 1] \ [ − n, n ] . There exists a polynomial g n +1 such that | h n +1 − g n +1 | < ǫ n +1 on E n +1 , h n +1 ( j ) = g n +1 ( j ) for j = − ( n + 1) , · · · , n + 1 and such that T ji ( h n +1 ) = T ji ( g n +1 ) , i = 1 , · · · , m ; j = − n, · · · , n + 1 . Thus, we have T ji ( f ) = T ji ( g n +1 ) , i = 1 , · · · , m ; j = − n, · · · , n + 1 ,f ( j ) = g n +1 ( j ) for j = − ( n + 1) , · · · , n + 1 . P. M. GAUTHIER AND J. KIENZLE and | f − g n +1 | < ǫ n +1 on [ − ( n + 1) , n + 1] \ [ − n, n ] ǫ n + ǫ n +1 on [ − n, n ] \ [ − ( n − , n − ...ǫ + ǫ + · · · + ǫ n +1 on [ − , | f − g n +1 | ≤ | f − h n +1 | + | h n +1 − g n +1 | = | h n +1 − g n +1 | < ǫ n +1 on [ − ( n + 1) , n + 1] \ [ − n, n ] | f − g n | + | h n +1 − g n +1 | < ǫ n + ǫ n +1 on [ − n, n ] \ [ − ( n − , n − ... | f − g n | + | h n +1 − g n +1 | < ǫ + ǫ + · · · + ǫ n +1 on [ − , . We also have that | g n +1 − g n | < ǫ n +1 on D n , since | g n +1 − g n | ≤ | g n +1 − h n +1 | + | h n +1 − g n | < ǫ n +1 on D n . Let us show that the sequence { g k } converges uniformly on compacta.It is sufficient to show that { g k } , is uniformly Cauchy on compactsubsets. For each k , we have that | g k − g k − | < ǫ k on D k − . Let K ⊂ C be an arbitrary compact set. For δ >
0, we choose N δ so large that K ⊂ D N δ and k > ℓ > N δ ⇒ P kj = ℓ ǫ j < δ . Then, for such k and ℓ , | g k − g ℓ | ≤ k − X j = ℓ | g j +1 − g j | ≤ k − X j = ℓ ǫ j +1 < δ , on K. Thus, the sequence g k converges uniformly on compacta. The limit g is therefore an entire function.Let us show that | f − g | < ǫ . Fix t ∈ R and choose ℓ = [ | t | ] + 1.Then, for all k ≥ ℓ, | f ( t ) − g k ( t ) | ≤ | f ( t ) − g m ( t ) | + k X j = ℓ +1 | g j ( t ) − g j − ( t ) | < k X j = ℓ ǫ j < ǫ ( t ) / . Now, we choose k ≥ ℓ so large that | g k ( t ) − g ( t ) | < ǫ ( t ) /
2. Then, | f ( t ) − g ( t ) | ≤ | f ( t ) − g k ( t ) | + | g ( t ) − g k ( t ) | < ǫ ( t ) . Finally, we must show that T ji ( g ) = T ji ( f ) , i = 1 , · · · , m ; j = Z .Fix j. For all k > | j | , we have j ∈ { ( k − , · · · , k } . Thus, by (1), T ji ( f ) = T ji ( g k ) , i = 1 , · · · , m PPROXIMATION BY ENTIRE FUNCTIONS 7 and consequently, T ji ( g ) = lim k →∞ T ji ( g k ) = lim k →∞ T ji ( f ) = T ji ( f ) . (cid:3) Proof of Theorem 3
Proof. : For simplicity, we shall prove Corollary 4, which is a specialcase of Theorem 3. The proof of the general theorem is an obviousmodification.We may assume that f ( i ) (0) = 0 , i = 0 , · · · , m and we may alsoassume that ǫ ( t ) = ǫ ( | t | ) and that ǫ ( | t | ) is decreasing, as | t | increases.Put ǫ o = ǫ, and for i = 1 , · · · , m ; put ǫ i ( t ) = ǫ i − ( | t | + 1) . Then, for i = 0 , · · · , m, we have ǫ i ( t ) = ǫ ( | t | ) , the functions ǫ ( t ) are decreasingas | t | increases and ǫ i > ǫ i +1 , i = 0 , · · · , m − . By Lemma 7 there exists an entire function g m such that g m (0) = f ( m ) (0); (cid:12)(cid:12)(cid:12) f ( m ) ( t ) − g m ( t ) (cid:12)(cid:12)(cid:12) < ǫ m ( t ); Z nn − f ( m ) ( t ) dt = Z nn − g m ( t ) dt ;and Z nn − Z x · · · Z x i f ( m ) ( t ) dtdx i · · · dx = Z nn − Z x · · · Z x i g m ( t ) dtdx i · · · dx ;for i = 1 , · · · , m −
1; and n ∈ Z . We define the following entire functions. g k ( z ) = Z z g k +1 ( ζ ) dζ ; k = m − , m − , · · · , . Thus, we have: g ′ k ( z ) = g k +1 ( z ) , k = 0 , · · · , m − R z g k +2 ( ζ ) dζ , k = 0 , · · · , m − . Hence, setting g = g , we have: g ′ ( z ) = g ( z ) , g ′′ ( z ) = g ( z ) , · · · g ( m ) ( z ) = g m ( z ) . Therefore (cid:12)(cid:12)(cid:12) f ( m ) ( x ) − g ( m ) ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) f ( m ) ( x ) − g m ( x ) (cid:12)(cid:12)(cid:12) < ǫ m ( x ) ≤ ǫ ( x ) . We shall now show that (cid:12)(cid:12)(cid:12) f ( m − ( x ) − g ( m − ( x ) (cid:12)(cid:12)(cid:12) < ǫ m − ( x ) ≤ ǫ ( x ) . P. M. GAUTHIER AND J. KIENZLE
Denoting the integer part of x by [ x ] , we have, if x ≥ (cid:12)(cid:12)(cid:12) f ( m − ( x ) − g ( m − ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z x [ f ( m ) ( t ) − g ( m ) ( t )] dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ x ] X n =1 Z nn − [ f ( m ) ( t ) − g ( m ) ( t )] dt + Z x [ x ] [ f ( m ) ( t ) − g ( m ) ( t )] dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x [ x ] [ f ( m ) ( t ) − g ( m ) ( t )] dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ m ([ x ]) = ǫ m − ([ x ] + 1) < ǫ m − ( x ) ≤ ǫ ( x ) . Similarly, if x ≤ , (cid:12)(cid:12)(cid:12) f ( m − ( x ) − g ( m − ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z x [ f ( m ) ( t ) − g ( m ) ( t )] dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ x ] X n = − Z nn +1 [ f ( m ) ( t ) − g ( m ) ( t )] dt + Z x [ x ] [ f ( m ) ( t ) − g ( m ) ( t )] dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x [ x ] [ f ( m ) ( t ) − g ( m ) ( t )] dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ m ([ x ]) = ǫ m − ([ x ] + 1) < ǫ m − ( x ) ≤ ǫ ( x ) . Next we show that | f ( m − ( x ) − g ( m − ( x ) | < ǫ m − ( x ) ≤ ǫ ( x ) . As inthe previous case, (cid:12)(cid:12)(cid:12) f ( m − ( x ) − g ( m − ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z x [ f ( m − ( x ) − g ( m − ( x )] dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x [ x ] [ f ( m − ( x ) − g ( m − ( x )] dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ m − ([ x ]) = ǫ m − ([ x ] + 1) < ǫ m − ( x ) ≤ ǫ ( x ) . Repeating the same argument m − | f ( i ) ( x ) − g ( i ) ( x ) | < ǫ ( x ) , x ∈ R and i = 0 , , · · · , m . (cid:3) References [1] Carleman, T. Sur un théorème de Weierstraß.
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