Strictly positive definite functions on the real line Fabrice Derrien To cite this version: Fabrice Derrien. Strictly positive definite functions on the real line. 2010. hal-00519325 HAL Id: hal-00519325 https://hal.archives-ouvertes.fr/hal-00519325 Preprint submitted on 20 Sep 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Strictly positive definite functions on the real line F. Derrien Univ Lille Nord de France F-59 000 LILLE, FRANCE UArtois, Laboratoire de Math´ematiquesde Lens EA 2462, F´ed´eration CNRS Nord-Pas-de-Calais FR 2956, F-62 300 LENS,FRANCE Abstract We give some necessary or sufficient conditions for a function to be strictly positive definite on R. This problem is intimately linked with the repartition of the zeros of trigonometric polynomials. Keywords: strictly positive definite functions, Bochner’s theorem, scattered data interpolation 2000 MSC: 42A82, 41A05, 41A15 (30C15, 42A75) 1. Introduction and notations 1.1. Introduction In the theory of scattered data interpolation with linear combinations of translates of a basis function, we have to deal with strictly positive definite functions rather than positive definite ones, or more generally with conditionally strictly positive definite functions [2, 4, 16]. This ensures that the interpolation problem is always solvable. Using Bochner’s characterization of continuous positive definite functions as Fourier transforms of nonnegative finite measures, it is straightfor- ward to see that verifying strict positive definiteness reduces to checking whether the exponen- tials are linearly independent on a certain subset of real numbers. Some years ago, K-F. Chang published a paper on this subject [3] but unfortunately his main result is erroneous. In fact, the PK iξkz author asserts in [3, theorem 3.5] that any nonzero complex polynomial t(z) = k=1 cke with ξk 2 R is of sine type that is there are positive constants a, b, σ and τ such that σjyj σjyj ae ≤ jt(x + iy)j ≤ be for all x; y 2 R; jyj ≥ τ: Taking for instance t(z) = eiz, we see easily that the first inequality can not be satisfied. As a consequence, the author concludes at the end of his paper that there does not exist nonzero 1 trigonometric polynomial vanishing at the points xn = n + 8 sgn (n) where n 2 Z, but trivially t(x) = sin(8πx) does. Email address: [email protected] (F. Derrien) Preprint submitted to Journal of Mathematical Analysis and Applications September 19, 2010 1.2. Notations Let us fix some usual notations for subspaces of complex valued functions defined on the real line. We denote by C the set of continuous functions and C0 its subspace of functions van- ishing at infinity, Ck the set of functions with k continuous derivatives, S the Schwartz space of infinitely differentiable and rapidly decreasing functions, D the space of infinitely differentiable functions with compact support, Lp the set of p-power integrable functions with respect to the Lebesgue measure λ, T the set of trigonometric polynomials that is the functions of the form PK iξk x t(x) = k=1 cke where ck are complex numbers and ξk real numbers, AP(R) the set of Bohr almost periodic functions. We will use also the notation AP(Z) for the set of complex valued Bohr almost periodic functions on Z. The set of real valued Bohr almost periodic functions on R, resp. Z, is denoted by APR(R), resp. APR(Z). For a function φ 2 C, we note φˇ, resp. φ˜, the function defined by φˇ(ξ) = φ(−ξ), resp. φ˜(ξ) = φ(−ξ). The symbol M+ stands for the set of nonnegative finite Borel measure on R. The support of a measure µ 2 M+ defined as the smallest closed subset of R whose complement has µ- measure 0 is denoted by supp µ. The Fourier transform of a measure µ 2 M+ is defined by R µˆ(ξ) = e−iξx dµ(x). R 2. Formulation of the interpolation problem, an analogue for Bochner’s theorem Given a function ' 2 C, an arbitrary set of distinct real numbers Ξ = fξ1; : : : ; ξKg and complex numbers z1;:::; zK, the scattered data interpolation problem consists in finding a function XK u(ξ) = ck'(ξ − ξk) k=1 such that u(ξ j) = z j; j = 1 ::: K. Although this problem is equivalent to the nonsingularity of the K × K matrix A with entries A jk = '(ξ j − ξk), one wishes to know when the interpolation matrix A is positive definite for any set Ξ. For this purpose, let us recall the following definition. Definition 2.1. A complex valued continuous function ' is said positive definite (resp. strictly positive definite) on R if for every finite set of distinct real numbers Ξ = fξ1; : : : ; ξKg and every K vector (c1;:::; cK) 2 C n f0g, the inequality XK XK c¯ jck'(ξ j − ξk) ≥ 0 (resp. > 0) (1) j=1 k=1 holds true. We denote by P (resp. Ps) the class of such functions. Clearly Ps ⊂ P. The class of positive definite functions is fully characterized by the Bochner’s theorem [1]. Theorem 2.1. A function ' 2 P if and only if ' = µˆ where µ 2 M+, ' and µ being biuniquely determined. Therefore we can ask for an equivalent characterization of a strictly positive definite function in terms of its Fourier transform. We state before a basic lemma. 2 Lemma 2.1. Let µ 2 M+ and f 2 C a nonnegative function. We have Z f dµ = 0 , f = 0 on supp µ. R Proof. Let us set X = supp µ. Since R n X is the largest open subset of R with µ-measure 0 and the set f f > 0g is open, we have Z f dµ = 0 , µ(f f > 0g) = 0 , f f > 0g ⊂ R n X , X ⊂ f f = 0g , f = 0 on X: R Proposition 2.1. Let ' = µˆ 2 P. Then ' 2 Ps if and only if there does not exist t 2 T n f0g vanishing on supp µ. PK iξk x Proof. Let t 2 T n f0g. Remark first that t can be written in the form t(x) = k=1 cke where xk are pairwise distinct real numbers and ck are complex numbers, not all zero. We have XK XK XK XK Z −i(ξ j−ξk)x c¯ jck'(ξ j − ξk) = c¯ jck e dµ(x) j=1 k=1 j=1 k=1 R Z XK XK −iξ j x iξk x = c¯ je cke dµ(x) R j=1 k=1 Z XK 2 = c eiξk x dµ(x) k R k=1 Z = jt(x)j2 dµ(x): R s From lemma 2.1 the last integral is 0, i.e. ' < P , if and only if t vanishes on supp µ. 3. Elementary properties of strictly positive definite functions The next result shows in particular that the class Ps is a convex cone, closed under multipli- cation. Theorem 3.1. Let ' 2 Ps and 2 P. Then (i) ' + 2 Ps, (ii) ' 2 Ps provided that , 0. Proof. Let t 2 T and write ' = µˆ, = νˆ so that ' + = µ[+ ν and ' = µ[ ? ν. It is clear that µ + ν and µ ? ν are in M+. We have from proposition 2.1 Z Z jtj2 d(µ + ν) = 0 =) jtj2 dµ = 0 =) t = 0: R R For the second assertion, we remark by Fubini’s theorem that Z Z Z Z jtj2 d(µ ? ν):= jt(x + y)j2 dµ(x) dν(y) = dν(y) jt(x + y)j2 dµ(x) 2 R R 3 R R and if we suppose , 0, then supp ν , ;. From lemma 2.1 and proposition 2.1, it follows that Z Z jtj2 d(µ ? ν) = 0 =) jt(x + y)j2 dµ(x) = 0 for y 2 supp ν =) t = 0: R R Theorem 3.2. Let ' be a function in Ps. Then (i) ' is hermitian i.e. 'ˇ = '¯, (ii) '(0) > 0 and j'(ξ)j < '(0) for all ξ , 0, (iii) '(a·) 2 Ps provided that a , 0, (iv) the functions ';¯ < ' and j'j2 are in Ps, s P1 n (v) h ◦ ' 2 P if h(z) = n=0 anz is a nonconstant power series with nonnegative coefficients, converging for z = '(0). R Proof. By theorem 2.1 we have ' = µˆ where µ 2 M+ so that '(−ξ) = eiξx dµ(x) = '(ξ). R In the definition 1, let n = 1 to obtain '(0) > 0 and then let n = 2; ξ1 = ξ; ξ2 = 0; c1 = 1; c2 = −1 which shows that '(0)2 − '(ξ)'(−ξ) = '(0)2 − j'(ξ)j2 > 0 whenever ξ , 0. The third assertion follows immediately from the definition 1. s 1 2 Taking a = −1 shows that' ¯ = 'ˇ 2 P and so are the functions <' = 2 (' + '¯) and j'j = ''¯ by the preceding theorem. n n P1 n P1 n Since jan' (ξ)j ≤ an' (0) for all ξ 2 R and all n ≥ 0 and n=0 an' (0) < 1, the series n=0 an' converges uniformly on R by the Weierstrass M-test and so its sum h◦' is continuous. Since h is n0 s nonconstant, there exists a coefficient an0 > 0 with n0 ≥ 1 and hence by theorem 3.1, an0 ' 2 P and each partial sum PN a 'n is either 0 or in Ps.