Space-time covariance functions with compact support

Spae-time o v ariane funtions with ompat supp ort Viktor P. Zastavnyi Donetsk National Universit y Depa rtment of Mathematis Universitetsk a y a str. 24, Donetsk, 340001, Ukraine zastavnrambler. ru Emilio Por u Universit y Jaume I of Castellón Depa rtment of Mathematis Campus Riu Se E-12071 Castellón, Spain porumat.uji.e s Abstrat We haraterize ompletely the Gneiting lass [6 ℄ of spae-time ovariane funtions and give more relaxed onditions on the involved funtions. We then show neessary onditions for the onstrution of ompatly supported funtions of the Gneiting type. These onditions are very general sine they do not depend on the Eulidean norm. Finally, we disuss a general lass of positive definite funtions, used for multivariate Gaussian random fields. For this lass, we show neessary riteria for its generator to be ompatly supported. Keywords : Compat support, Gneiting's lass, Positive definite, Spae-time. Spae-time o v ariane funtions with ompat supp ort Viktor P . Zasta vn yi and Emilio P oru 1 In tro dution Reen t literature p ersisten tly emphasizes the use of appro ximation metho ds and new metho dologies for dealing with massiv e spatial data set. When dealing with spatial data, alulation of the in v erse of the o v ariane matrix b eomes a ruial problem. F or instane, the in v erse is needed for b est linear un biased predition (alias kriging), and is rep eatedly alulated in the maxim um lik eliho o d estimation or the Ba y esian inferenes. Th us, large spatial sample sizes tradue in to big  hallenges from the omputational p oin t of view. A natural idea that made proselytes in the last y ear is to mak e the o v arianes exatly zero after ertain distane so that the resulting matrix has a high prop ortion of zero en tries and is therefore a sparse matrix. Op erations on sparse matries tak e up less omputer memories and run faster. Ho w ev er, this should b e done in a w a y to preserv e p ositiv e deniteness of the resulting o v ariane matrix. The idea go es under the name of o v ariane tap ering , b y meaning that the true o v ariane is m ultiplied p oin t wise with a ompatly supp orted and radial orrelation funtion. This op eration is te hnially justied b y the fat that the S h ur pro dut preserv es p ositiv e deniteness. The eets of tap ering in terms of estimation and in terp olation ha v e b een reen tly insp eted b y [4 ℄, where general onditions are giv en in order to ensure that tap ering do es not aet the eieny of the maxim um lik eliho o d estimator. F or spatial in terp olation, [ 5℄ sho w that under some regularit y onditions, tap ering pro edures yield asymptotially optimal predition. In order to assess these prop erties, the asymptoti framew ork adopted b y the authors is of the inll t yp e, and the to ol allo wing to ev aluate the p erformanes of tap ering is the equiv alene of Gaussian measures, for whi h a omprehensiv e theory an b e found in the seminal w ork b y Y adrenk o [13 ℄. These p oin ts x v ery briey the state of the art and w e refer the reader to [ 3 ℄ for an exellen t surv ey on the topi. Although tap ering has b een w ell understo o d in the spatial framew ork, there is nothing done, to the kno wledge of the authors, for the spatio-temp oral ase. In partiular, the use of tap ering is at least questionable in spae-time, sine the same t yp e of asymptotis do es not apply and th us it is not easy to ev aluate its p erformanes. But a deep er lo ok at this problem also highligh ts the non existene, in the literature, of spae-time o v ariane funtions that are ompatly supp orted o v er spae, time or b oth. These fats motiv ate the resear h do umen ted 1 in this man usript. W e deal with  hallenges related to spae-time o v ariane funtions. If spatial data set an b e massiv e, one an imagine ho w the dimensionalit y problem aets spae-time estimation and in terp olation. This problem ma y b e faed on the base of t w o p ersp etiv es that an b e illustrated through the elebrated T. Gneiting lass of o v ariane funtions [6℄: for ( x, t ) ∈ R d + l , the funtion ( x, y ) 7→ K ( x, t ) := h ( k t k 2 ) − d/ 2 ϕ  k x k 2 h ( k t k 2 )  (1) is p ositiv e denite, for ϕ ompletely monotone on the p ositiv e real line and h a Bernstein funtion. F or l = 1 , the funtion ab o v e is a stationary and nonseparable spae-time o v ariane. This funtion has b een p ersisten tly used b y the literature and a Go ogle s holar sear h highligh ts that urren tly there are o v er 90 pap ers where this o v ariane has b een used for appliations to spae-time data. If there are man y observ ations o v er spae, time or b oth, then the use of this funtion w ould b e questionable for the omputational reasons exp osed ab o v e. A more in triguing p ersp etiv e is to onsider a funtion of the Gneiting t yp e, but replaing the generator ϕ in equation (1) with a ompatly supp orted funtion, and insp eting the onditions ensuring that p ermissibilit y is preserv ed on some d -dimensional Eulidean spae. The results are illustrated in the follo wing setions. An auxiliary result of indep enden t in terest is also giv en: w e  haraterize ompletely the Gneiting lass and giv e more general onditions for its p ermissibilit y . The ratio men tis of this pap er leads then to onsider a general lass of o v arianes, originally prop osed in P oru et al. [10 ℄ and more reen tly in [1℄. Both groups of authors sho w that is lass of o v arianes an b e v ery v ersatile sine it an b e used for t w o-fold purp oses: on the one hand, it an b e eetiv ely used to deal with zonally anisotropi strutures, on the other hand it an b e adapted to represen t the o v ariane mapping asso iated to a m ultiv ariate random eld, whi h is highly in demand sine there are v ery few mo dels with these  harateristis [ 7 ℄. As a onlusion to the preludium, the plan of the pap er is the follo wing: in Setion 2 w e presen t basi fats ab out p ositiv e and negativ e denite funtions. Setion 3  haraterizes ompletely the Gneiting lass, for whi h only suien t onditions w ere kno wn un til no w. In Setion 4 w e presen t neessary onditions for ompatly supp orted o v arianes of the Gneiting t yp e. Similar results are obtained in Setion 5 for the m ultiv ariate lass of ross-o v arianes prop osed in [10 ℄. 2 Preliminaries This setion is largely exp ository and on tains basi fats and information needed for a self-on tained exp osition. W e shall en uniate the onepts of p ositiv e and negativ e deniteness, as w ell as the material related to them, 2 w orking with linear spaes and subspaes. The spae-time notation will b e used only when neessary for a learer exp osition of results. F or E a real linear spae, w e denote b y FD ( E ) the set of all linear nite-dimensional subspaes of E . If dim E = n ∈ N and e 1 , . . . , e n are basis in E , then f ∈ C ( E ) ⇐ ⇒ f ( x 1 e 1 + . . . + x n e n ) ∈ C ( R n ) and f ∈ L ( E ) ⇐ ⇒ f ( x 1 e 1 + . . . + x n e n ) ∈ L ( R n ) . Also, w e all C 0 ( E ) the set of all funtion f ∈ C ( E ) su h that f has ompat supp ort. If dim E = ∞ , then f ∈ C ( E ) ⇐ ⇒ f ∈ C ( E 0 ) ∀ E 0 ∈ FD( E ) . A omplex-v alued funtion f : E → C is said to b e p ositiv e denite on E (denoted hereafter f ∈ Φ( E ) ) if for an y nite olletion of p oin ts { ξ i } n i =1 ∈ E the matrix ( f ( ξ i − ξ j )) n i,j =1 is p ositiv e denite, i.e. for all a 1 , a 2 , . . . , a n ∈ C : n X i,j =1 a i f ( ξ i − ξ j ) a j ≥ 0 . It is w ell kno wn that the family of p ositiv e denite funtions is a on v ex one whi h is losed under addition, pro duts, p oin t wise on v ergene and sale mixtures. Briey , w e ha v e the follo wing prop erties. Let f , f i ∈ Φ( E ) , i ∈ N . Then: 1. | f ( x ) | ≤ f (0) , f ( − x ) = f ( x ) , | f ( x ) − f ( h ) | 2 ≤ 2 f ( 0)Re( f (0) − f ( x − h )) , x , h ∈ E ; 2. λ 1 f 1 + λ 2 f 2 with λ i ≥ 0 , ¯ f , Re f , f 1 f 2 ∈ Φ( E ) ; 3. if, for all x ∈ E , the nite limit lim n →∞ f n ( x ) =: g ( x ) exists, then g ∈ Φ( E ) ; 4. for an y linear op erator A : E 1 → E the funtion f ◦ A b elongs to Φ( E 1 ) ; in partiular, f ∈ Φ( E 1 ) for an y linear subspae E 1 from E . Let E = R n . The elebrated Bo  hner's theorem establishes a one to one orresp ondene b et w een on tin uous p ositiv e denite funtions and the F ourier transform of a p ositiv e and b ounded measure, i.e. f ( x ) = F n ( µ ( u ))( x ) . If µ is absolutely on tin uous with resp et to the Leb esgue measure, than dµ ( u ) = b f ( u ) du , for b f nonnegativ e. This an b e rephrased in the follo wing w a y: if f ∈ C ( R n ) ∩ L ( R n ) , then f ∈ Φ( R n ) if and only if b f ( u ) = F − 1 n ( f )( u ) := Z R n e i ( u,x ) f ( x ) dx ≥ 0 , u ∈ R n , for ( · , · ) the usual dot pro dut. The funtion b f is alled sp etral densit y or F ourier pair asso iated to f . 3 If f is a radially symmetri and on tin uous funtion dep ending on the squared Eulidean norm k · k 2 2 , i.e. f ( x ) = ϕ ( k x k 2 2 ) , ϕ ∈ C [0 , + ∞ ) , then the F ourier transform ab o v e simplies to the Bessel in tegral (if in addition f ∈ L ( R n ) ) g n ( s ) := Z + ∞ 0 ϕ ( u 2 ) u n − 1 j n 2 − 1 ( su ) du , (2) where j λ ( u ) := J λ ( u ) u λ , with J λ a Bessel funtion of the rst kind. Th us f ∈ Φ( R n ) , for some n ∈ N and for f radially symmetri, if and only if g n ( u ) ≥ 0 ∀ u > 0 . A funtion f :]0 , ∞ [ → R is alled  ompletely monotone , if it is arbitrarily often dieren tiable and ( − 1) n f ( n ) ( x ) ≥ 0 for x > 0 , n = 0 , 1 , . . . . By Bernstein's theorem the set M (0 , ∞ ) of ompletely monotone funtions oinides with that of Laplae transforms of p ositiv e measures µ on [0 , ∞ [ , i.e. f ( x ) = L µ ( x ) = Z [0 , ∞ [ e − xt dµ ( t ) , where w e only require that e − xt is µ -in tegrable for an y x > 0 . M (0 , ∞ ) is a on v ex one whi h is losed under addition, m ultipliation and p oin t wise on v ergene. The onnetion with the funtion g n ( · ) giv es the elebrated S ho en b erg (1939) theorem b y whi h a radial funtion f ( x ) = ϕ ( k x k 2 2 ) , ϕ ∈ C [0 , + ∞ ) , b elongs to Φ( R n ) for all n ∈ N if and only if ϕ is ompletely monotone on the p ositiv e real line, and in this ase the Bessel in tegral in equation ( 2) redues to a Gaussian mixture. Finally , a Bernstein funtion is a p ositiv e funtion that is innitely often dieren tiable and whose rst deriv ativ e is ompletely monotone. F or a more detailed exp osition on these fats the reader is referred to [11℄. In this pap er w e shall b e also dealing with funtions dep ending not on the Eulidean norm but on some homogeneous on tin uous funtion ρ : E → R su h that ρ ( tx ) = | t | ρ ( x ) ∀ t ∈ R , x ∈ E and ρ ( x ) > 0 , x 6 = 0 . If ϕ ∈ C [0 , + ∞ ) and R + ∞ 0 | ϕ ( t 2 ) | t n − 1 dt < + ∞ , then w e ha v e that ϕ ◦ ρ 2 ∈ Φ( R n ) if and only if the funtion R n ∋ v 7→ G n ( v ) := Z R n ϕ ( ρ 2 ( y )) e i ( y, v ) dy (3) is nonnegativ e for all v ∈ R n . If ρ is the Eulidean norm, then the funtions G n ( · ) and g n ( · ) are related b y the w ell kno w equalit y G n ( v ) = (2 π ) n 2 g n ( || v || 2 ) . Finally , a omplex-v alued funtion h : E → C is alled (onditionally) negativ e denite on E (denoted h ∈ N ( E ) hereafter) if the inequalit y n X k,j =1 c k ¯ c j h ( x k − x j ) ≤ 0 is satised for an y nite systems of omplex n um b ers c 1 , c 2 , ..., c n , P n k =1 c k = 0 , and p oin ts x 1 , ..., x n in E . 4 Let { Z ( ξ ) , ξ ∈ R n } b e a on tin uous w eakly stationary and Gaussian random eld (RF for short). The asso iated o v ariane funtion f : R n → R is p ositiv e denite. This an b e rephrased b y sa ying that p ositiv e deniteness of a andidate on tin uous funtion f : R n → R is suien t ondition for the existene of a on tin uous w eakly stationary and Gaussian RF ha ving f ( · ) as o v ariane funtion. If, additionally , f ( · ) is r adial ly symmetri , the asso iated Gaussian RF is alled isotr opi . Isotrop y and sta- tionarit y are indep enden t assumptions but throughout the pap er w e shall assume b oth in order to k eep things simple. T o omplete the piture, the v ariane of the inremen ts of an in trinsially stationary Gaussian RF is alled v ariogram. F or t w o p oin ts of R n , sa y ξ i , i = 1 , 2 , w e ha v e that V ar ( Z ( ξ 2 ) − Z ( ξ 1 )) := γ ( ξ 2 − ξ 1 ) . The mapping γ ( · ) : R n → R is onditionally negativ e denite. The additional prop ert y of isotrop y is then analogously dened as b efore. 3 Complete Charaterization of the Gneiting lass Lemma 1. i. f ∈ Φ( E ) ⇐ ⇒ f ∈ Φ( E 0 ) ∀ E 0 ∈ FD( E ) . ii. If dim E = n ∈ N then f ∈ Φ( E ) ⇐ ⇒ f g ∈ Φ( E ) ∀ g ∈ Φ( E ) ∩ C 0 ( E ) . Pr o of. i. The neessit y is ob vious. As for the suieny , for n ∈ N and x 1 , ..., x n in E , w e ha v e that x 1 , ..., x n ∈ E 0 - the linear span of these elemen ts. Ob viously dim E 0 ≤ n . ii. Again, the neessit y is ob vious. F or the suieny , let e 1 , . . . , e n b e basis in E . Then w e tak e g ( x 1 e 1 + . . . + x n e n ) = (1 − ε | x 1 | ) + · . . . · (1 − ε | x n | ) + and ε ↓ 0 . The pro of is ompleted. Lemma 2. L et the next  onditions b e satise d: 1. h, b ∈ C ( E ) and h ( t ) > 0 ∀ t ∈ E . 2. ϕ ∈ C [0 , + ∞ ) and for the some m ∈ N : R + ∞ 0 | ϕ ( u 2 ) | u m − 1 du < + ∞ . 3. ρ ∈ C ( R m ) , ρ ( tx ) = | t | ρ ( x ) ∀ t ∈ R , x ∈ R m and ρ ( x ) > 0 , x 6 = 0 . Then K ( x, t ) := b ( t ) ϕ  ρ 2 ( x ) h ( t )  ∈ Φ( R m × E ) ⇐ ⇒ b ( t )( h ( t )) m 2 G m ( p h ( t ) v ) ∈ Φ( E ) ∀ v ∈ R m , 5 with G m ( · ) dene d in e quation (3). Pr o of. Observ e that ϕ  ρ 2 ( x )  ∈ L ( R m ) . W e ha v e that K ( x, t ) ∈ Φ ( R m × E ) ⇐ ⇒ K ( x, t ) ∈ Φ( R m × E 0 ) ∀ E 0 ∈ FD( E ) ⇐ ⇒ K ( x, t ) g ( t ) ∈ Φ( R m × E 0 ) ∀ E 0 ∈ FD( E ) , ∀ g ∈ Φ( E 0 ) ∩ C 0 ( E 0 ) ⇐ ⇒ Z Z R m × E 0 K ( x, t ) g ( t ) e i ( x,v ) e i ( t,u ) dxdt ≥ 0 ∀ E 0 ∈ FD( E ) , ∀ g ∈ Φ( E 0 ) ∩ C 0 ( E 0 ) , ∀ v ∈ R m , u ∈ E 0 . As for the last in tegral, a  hange of v ariables of the t yp e x = p h ( t ) y yields that the last inequalit y is equiv alen t to Z E 0 g ( t ) b ( t )( h ( t )) m 2 G m ( p h ( t ) v ) e i ( t,u ) dt ≥ 0 , ∀ v ∈ R m , u ∈ E 0 , whi h holds if, and only if ∀ g ∈ Φ( E 0 ) ∩ C 0 ( E 0 ) , ∀ v ∈ R m , w e ha v e g ( t ) b ( t )( h ( t )) m 2 G m ( p h ( t ) v ) ∈ Φ( E 0 ) ∀ E 0 ∈ FD( E ) , ⇐ ⇒ b ( t )( h ( t )) m 2 G m ( p h ( t ) v ) ∈ Φ( E ) ∀ v ∈ R m . The pro of is ompleted. The follo wing result giv es a omplete  haraterization of the Gneiting lass, with the additional feature that only negativ e deniteness of the funtion h is required, whilst Gneiting's assumptions are m u h more restritiv e as it is required that h ′ is ompletely monotone on the p ositiv e real line. F urthermore, w e giv e a simple pro of of this result and w e defer it to next setion for the reasons that will b eome apparen t throughout the pap er. Theorem 3. L et h ∈ C ( E ) , h ( t ) > 0 ∀ t ∈ E . L et d ∈ N . The fol lowing statements ar e e quivalent: 1. K ( x, t ) := ( h ( t )) − d 2 ϕ  || x || 2 2 h ( t )  ∈ Φ( R d × E ) ∀ ϕ ∈ C [0 , + ∞ ) T M (0 , + ∞ ) . 2. e − λh ( t ) ∈ Φ( E ) ∀ λ > 0 . Let us onsider examples of funtions h for whi h the statemen t 2 in Theorem 3 holds. Example 1 Let h ( t ) = || t || α p + c , c > 0 , 0 < p ≤ + ∞ , α ≥ 0 , t = ( t 1 , . . . , t n ) ∈ R n , where || t || p p = P n k =1 | t k | p , 0 < p < ∞ , and || t || ∞ = sup 1 ≤ k ≤ n | t k | . Then e − λh ( t ) ∈ Φ( R n ) ∀ λ > 0 ⇐ ⇒ e −|| t || α p ∈ Φ( R n ) ⇐ ⇒ 0 ≤ α ≤ α ( l n p ) , where α ( l n p ) =        2 if n = 1 , 0 < p ≤ ∞ ; p if n ≥ 2 , 0 < p ≤ 2; 1 if n = 2 , 2 < p ≤ ∞ ; 0 if n ≥ 3 , 2 < p ≤ ∞ . (4) 6 F or 0 < p ≤ 2 , w e get S ho en b erg's result. The other t w o ases ha v e b een in v estigated b y K oldobsky [8 ℄ in 1991 and Zasta vn yi [14 , 15 , 16 ℄ in 1991 ( 2 < p ≤ ∞ , n ≥ 2 ). Finally , Misiewiez [ 9 ℄ ga v e the last result in 1989 ( p = ∞ , n ≥ 3 ). Example 2 If ρ ( t ) is a norm on R 2 , then e − ρ α ( t ) ∈ Φ( R 2 ) for all 0 ≤ α ≤ 1 . This is a w ell-kno w fat (see, for example, [17 ℄). Therefore e − λh ( t ) ∈ Φ( R 2 ) ∀ λ > 0 , where h ( t ) = ρ α ( t ) + c , 0 ≤ α ≤ 1 , c > 0 . Example 3 Let ψ ( s ) ∈ R ∀ s > 0 . Then, it is w ell kno wn that e − λψ ∈ M (0 , + ∞ ) ∀ λ > 0 ⇐ ⇒ ψ ′ ∈ M (0 , + ∞ ) . Gneiting [6 ℄ pro v es the follo wing: if ψ ∈ C [0 , + ∞ ) , ψ ( s ) > 0 ∀ s ≥ 0 , and ψ ′ ∈ M (0 , + ∞ ) , then e − λh ( t ) ∈ Φ( R n ) for all λ > 0 , n ∈ N , where h ( t ) := ψ ( || t || 2 2 ) and, hene, K ( x, t ) := ( ψ ( || t || 2 2 )) − d 2 ϕ  || x || 2 2 ψ ( || t || 2 2 )  ∈ Φ( R d × R n ) ∀ ϕ ∈ C [0 , + ∞ ) \ M (0 , + ∞ ) , d ∈ N . Example 4 By elebrated S ho en b erg's Theorem [12 ℄, if h ( − t ) = h ( t ) ∀ t ∈ E , then h ( t ) ∈ N ( E ) ⇐ ⇒ e − λh ( t ) ∈ Φ( E ) ∀ λ > 0 . Example 5 Let g ∈ Φ( E ) , g ( − t ) = g ( t ) for all t ∈ E and h ( t ) := g (0) − g ( t ) + c , c > 0 . Then h ( t ) > 0 ∀ t ∈ E , h ∈ N ( E ) and, hene, e − λh ( t ) ∈ Φ( E ) for all λ > 0 . 4 Neessary onditions for ompatly supp orted funtions of the Gneit- ing t yp e F rom no w on let us write S d − 1 := { x ∈ R d : || x || 2 = 1 } for the sphere of R d . Theorem 4. L et the next  onditions b e satise d: 1) h ∈ C ( E ) , h ( t ) > 0 ∀ t ∈ E and h ( t ) 6≡ h (0) on E . 2) ϕ ∈ C [0 , + ∞ ) , ϕ (0) > 0 . 7 3) F or d ∈ N , ρ ∈ C ( R d ) , ρ ( tx ) = | t | ρ ( x ) ∀ t ∈ R , x ∈ R d and ρ ( x ) > 0 , x 6 = 0 . 4) K ( x, t ) := ( h ( t )) − d 2 ϕ  ρ 2 ( x ) h ( t )  ∈ Φ( R d × E ) . Then: 1. ( h ( t )) − d 2 ∈ Φ( E ) and ϕ  ρ 2 ( x )  ∈ Φ( R d ) . 2. If ther e exists a n ∈ N T [1 , d ] suh that R + ∞ 0 | ϕ ( u 2 ) | u n − 1 du < + ∞ , then ∀ m = 1 , . . . , n and v ∈ R m the funtion s 7→ f m,v ( s ) := s m − d G m ( sv ) , with G m ( · ) as dene d in (3 ), is de r e asing on (0 , + ∞ ) . F urthermor e, f m,v (+ ∞ ) = 0 for v 6 = 0 . 3. If R + ∞ 0 | ϕ ( u 2 ) | u d − 1 du < + ∞ , then G d (0) > 0 . If, in addition, G d is r e al-analyti, then ∀ v ∈ R d , v 6 = 0 the funtion s 7→ f d,v ( s ) := G d ( sv ) is stritly de r e asing on [0 , + ∞ ) and G d ( v ) > 0 ∀ v ∈ R d . 4. If R + ∞ 0 | ϕ ( u 2 ) | u d +1 du < + ∞ , then α 1 ( v ) := R R d ϕ ( ρ 2 ( y ))( y , v ) 2 dy ≥ 0 ∀ v ∈ S d − 1 and β 1 := R R d ϕ ( ρ 2 ( y )) || y || 2 2 dy ≥ 0 . F urthermor e, α 1 ( v ) ≡ 0 on S d − 1 ⇐ ⇒ β 1 = 0 . If, in addition, β 1 > 0 , then e − λh ( t ) ∈ Φ( E ) ∀ λ > 0 . 5. If R + ∞ 0 | ϕ ( u 2 ) | e εu du < + ∞ for some ε > 0 (for example, when ϕ has  omp at supp ort), then ∃ p ∈ N : e − λh p ( t ) ∈ Φ( E ) ∀ λ > 0 . In pr ati e, for p it is p ossible to take one of the fol lowing numb ers: p ( v ) := min  k ∈ N : α k ( v ) = Z R d ϕ ( ρ 2 ( y ))( y , v ) 2 k dy 6 = 0  , v ∈ S d − 1 , q := min  k ∈ N : β k = Z R d ϕ ( ρ 2 ( y )) || y || 2 k 2 dy 6 = 0  . The funtion p ( · ) is b ounde d on S d − 1 and q = min v ∈ S d − 1 p ( v ) . Pr o of. The statemen t 1 is ob vious. Let us pro v e the statemen t 2. By Lemma 2, w e ha v e F m,v ( t ) := ( h ( t )) m − d 2 G m ( p h ( t ) v ) ∈ Φ( E ) , ∀ m = 1 , n , v ∈ R m . Hene, F m,v (0) = ( h (0)) m − d 2 G m ( p h (0) v ) ≥ 0 and | F m,v ( t ) | ≤ F m,v (0) , t ∈ E . Therefore G m ( v ) ≥ 0 , v ∈ R m , and ( sh ( t )) m − d 2 G m ( p h ( t ) sv ) ≤ ( sh (0)) m − d 2 G m ( p h (0) sv ) , ∀ m = 1 , n , v ∈ R m , s > 0 , t ∈ E . The latter inequalit y is equiv alen t to f m,v s h ( t ) h (0) · s ! ≤ f m,v ( s ) , ∀ m = 1 , n , v ∈ R m , s > 0 , t ∈ E . Sine ( h ( t )) − d 2 ∈ Φ( E ) , then h ( t ) ≥ h (0) , t ∈ E . Sine h ( t ) 6≡ h (0) on E , then there exists a p oin t t 0 ∈ E su h that q := q h ( t 0 ) h (0) > 1 . By the in termediate v alues Theorem ∀ α ∈ [1 , q ] ∃ ξ ∈ E : q h ( ξ ) h (0) = α . Therefore, 8 f m,v ( αs ) ≤ f m,v ( s ) for all s > 0 and α ∈ [1 , q ] . Hene, f m,v ( α 2 s ) ≤ f m,v ( αs ) ≤ f m,v ( s ) for all s > 0 and α ∈ [1 , q ] . Th us, f m,v ( α p s ) ≤ f m,v ( s ) for all s > 0 , α ∈ [1 , q ] and p ∈ N . This implies that the funtion f m,v ( s ) dereases in s ∈ (0 , + ∞ ) . By the Riemann-Leb esgue Theorem, it follo ws that G m ( v ) → 0 as || v || 2 → + ∞ . Hene f m,v (+ ∞ ) = 0 for v 6 = 0 . The statemen t 2 is pro v ed. Let us pro v e the statemen t 3. i . F rom statemen t 2 it follo ws that for all v ∈ R d , v 6 = 0 , the funtion G d ( sv ) dereases in s ∈ [0 , + ∞ ) and, hene, 0 ≤ G d ( v ) ≤ G d (0) . Therefore, G d (0) > 0 (otherwise G d ( v ) ≡ 0 on R d ⇒ ϕ ( ρ 2 ( y )) ≡ 0 on R d , that on tradits the ondition ϕ (0) > 0 ). ii . If, in addition, G d is real-analyti, then ∀ v ∈ R d , v 6 = 0 , the funtion G d ( sv ) stritly dereases on [0 , + ∞ ) . This an b e pro v ed b y on traddition. Let us assume that, for some v 0 ∈ R d and v 0 6 = 0 , the funtion G d ( sv 0 ) is onstan t on some in terv al ( α, β ) ⊂ (0 , + ∞ ) , α < β . This w ould imply that G d it is onstan t on [0 , + ∞ ) and G d (0) = lim s → + ∞ G d ( sv 0 ) = 0 , whi h on tradits i . Th us, ∀ v ∈ R d , v 6 = 0 , the funtion G d ( sv ) stritly dereases on [0 , + ∞ ) and, hene, G d ( v ) > lim s → + ∞ G d ( sv ) = 0 . The statemen t 3 is pro v ed. Let us pro v e statemen t 4. Let v ∈ S d − 1 and f d,v ( s ) := G d ( sv ) . F rom statemen ts 2 and 3 , it follo ws that the funtion f d,v ( s ) dereases on [0 , + ∞ ) and that f d,v (0) > 0 . Ob viously , f d,v ( s ) ∈ C 2 ( R ) and f d,v ( s ) = f d,v (0) + f ′′ d,v (0) 2 s 2 + o ( s 2 ) , s → 0 , where f ′′ d,v (0) = − α 1 ( v ) . Note that f ′′ d,v (0) ≤ 0 , otherwise the funtion f d,v ( s ) strongly inreases on [0 , c ] for some c > 0 , whi h on tradits statemen t 2 . Th us, α 1 ( v ) ≥ 0 for all v ∈ S d − 1 . F or p > 0 , the next in tegral is onstan t on S d − 1 : Z S d − 1 | ( y , v ) | p dσ ( v ) ≡ c d,p > 0 , y ∈ S d − 1 , where dσ , if n ≥ 2 , is the surfae measure on S d − 1 and dσ ( v ) = δ ( v − 1) + δ ( v + 1) , if d = 1 (here δ ( v ) - the Dira measure with mass 1 onen trated in the p oin t v = 0 ). Therefore, Z S d − 1 | ( y , v ) | p dσ ( v ) = c d,p || y || p 2 , y ∈ R d , p > 0 . (5) Hene Z S d − 1 α 1 ( v ) dσ ( v ) = c d, 2 β 1 ≥ 0 and α 1 ( v ) ≡ 0 on S d − 1 ⇐ ⇒ β 1 = 0 . Let, in addition, β 1 > 0 . Then f ′′ d,v 0 (0) = − α 1 ( v 0 ) < 0 for some v 0 ∈ S d − 1 and ψ n ( t ) := G d ( γ n p h ( t ) v 0 ) G d (0) ! n = (1 + g n ( t )) n ∈ Φ( E ) , ∀ n ∈ N , γ n > 0 . (6) T ak e γ n := − 2 f d,v 0 (0) f ′′ d,v 0 (0) · λ n ! 1 2 > 0 , λ > 0 . 9 Ob viously , γ n → +0 and g n ( t ) = f d,v 0 ( γ n p h ( t )) − f d,v 0 (0) f d,v 0 (0) ∼ f ′′ d,v 0 (0) 2 f d,v 0 (0) · ( γ n p h ( t )) 2 = − λ n · h ( t ) , n → ∞ . Therefore, ψ n ( t ) → e − λh ( t ) and, hene, e − λh ( t ) ∈ Φ( E ) for all λ > 0 . The statemen t 4 is pro v ed. Let us pro v e the statemen t 5. In this ase G d is real-analyti and f (2 k ) d,v (0) = ( − 1) k α k ( v ) , f (2 k − 1) d,v (0) = 0 , Z S d − 1 α k ( v ) dσ ( v ) = c d, 2 k β k , k ∈ N . (7) Therefore, ∀ v ∈ S d − 1 ∃ p ∈ N so that f d,v ( s ) = f d,v (0) + f (2 p ) d,v (0) (2 p )! s 2 p + o ( s 2 p ) , s → 0 , where f (2 p ) d,v (0) 6 = 0 , otherwise the funtion f d,v (0) ≡ f d,v ( s ) ≡ f d,v (+ ∞ ) = 0 whi h on tradits the inequalit y G d (0) > 0 (see statemen t 3 ). Hene, f (2 p ) d,v (0) < 0 , otherwise the funtion f d,v ( s ) strongly inreases on [0 , c ] for some c > 0 , whi h on tradits statemen t 2 . Th us the funtion p ( v ) , v ∈ S d − 1 , denes orretly . Let v ∈ S d − 1 and p = p ( v ) . T ak e funtion (6), where v 0 = v γ n := − (2 p )! f d,v 0 (0) f (2 p ) d,v 0 (0) · λ n ! 1 2 p > 0 , λ > 0 . Then g n ( t ) ∼ − λ n · h p ( t ) , n → ∞ . Therefore ψ n ( t ) → e − λh p ( t ) and, hene, e − λh p ( t ) ∈ Φ( E ) for all λ > 0 . If α k ( v 0 ) 6 = 0 for some v 0 ∈ S d − 2 , k ∈ N , then α k ( v ) 6 = 0 in some neigh b orho o d of a p oin t v 0 and, hene, p ( v ) ≤ p ( v 0 ) in this neigh b orho o d. Th us the funtion p ( v ) is lo ally b ounded on ompat S d − 1 and, hene, p ( v ) is b ounded on S d − 1 . Let m = min v ∈ S d − 1 p ( v ) = p ( v 0 ) for some v 0 ∈ S d − 1 . Then α m ( v 0 ) 6 = 0 and for all v ∈ S d − 1 equalit y f d,v ( s ) = f d,v (0) + f (2 m ) d,v (0) (2 m )! s 2 m + o ( s 2 m ) , s → 0 holds. Ob viously ( − 1) k α k ( v ) = f (2 k ) d,v (0) = 0 , for all 1 ≤ k < m (if m ≥ 2 ), and ( − 1) m α m ( v ) = f (2 m ) d,v (0) ≤ 0 (otherwise the funtion f d,v ( s ) strongly inreases on [0 , c ] for some c > 0 that on tradits a statemen t 2 ). F rom (7) follo ws that β k = 0 for all 1 ≤ k < m (if m ≥ 2 ) and ( − 1) m β m < 0 . Therefore q = m . The Theorem 4 is pro v ed. Pro of of Theorem 3 . If h ( t ) ≡ h (0) > 0 on E , then the impliation 1) ⇒ 2) is ob vious. If h ( t ) 6≡ h (0) on E , then this impliation follo ws from statemen t 4 of Theorem 4 for ϕ ( s ) = e − s ∈ C [0 , + ∞ ) T M (0 , + ∞ ) . The rev erse impliation 2) ⇒ 1) follo ws from Lemma 2 for ϕ ( s ) = e − s , equalit y Z R d e − 1 2 σ || y || 2 2 e i ( y, v ) dy = (2 π σ ) d 2 e − σ 2 || v || 2 2 , v ∈ R d , σ > 0 , and Bernstein-Widder's Theorem. The pro of is ompleted. 10 Next Theorem 5 is an addition to Theorem 4 for the ase ρ ( x ) = || x || 2 . Theorem 5. L et the next  onditions b e satise d: 1) h ∈ C ( E ) , h ( t ) > 0 ∀ t ∈ E and h ( t ) 6≡ h (0) on E . 2) ϕ ∈ C [0 , + ∞ ) , ϕ (0) > 0 . 3) K ( x, t ) := ( h ( t )) − d 2 ϕ  || x || 2 2 h ( t )  ∈ Φ( R d × E ) . If R + ∞ 0 | ϕ ( u 2 ) | u m − 1 du < + ∞ for some natur al m ∈ [1 , d ] and g m is r e al-analyti, then the funtion f m ( s ) := s m − d g m ( s ) stritly de r e ases on (0 , + ∞ ) and g m ( s ) > 0 for al l s > 0 . Pr o of. F rom Theorem 4 it follo ws that f m dereases on (0 , + ∞ ) and f m ( s ) ≥ f m (+ ∞ ) = 0 for s > 0 . Sine f m is real-analyti on (0 , + ∞ ) , then funtion f m ( s ) stritly dereases on (0 , + ∞ ) . Otherwise the funtion f m is onstan t on some in terv al ( α, β ) ⊂ (0 , + ∞ ) , α < β , and, hene, it is onstan t on (0 , + ∞ ) and f m ( s ) = f m (+ ∞ ) = 0 , s > 0 . Therefore, G m ( v ) = (2 π ) m 2 g m ( || v || 2 ) ≡ 0 on R m . Hene, ϕ ( || x || 2 2 ) ≡ 0 on R m , whi h on tradits the ondition ϕ (0) > 0 . Th us, the funtion f m stritly dereases on (0 , + ∞ ) and, hene, f m ( s ) > f m (+ ∞ ) = 0 for all s > 0 . The Theorem 5 is pro v ed. 5 Some statemen ts in v olving a v ersatile general o v ariane funtion Previous results an b e generalized to the lass of p ositiv e denite funtions built in [ 10 ℄ and used for the purp oses highligh ted in Setion 1. Lemma 6. L et ϕ ∈ C ([0 , + ∞ ) n ) , n ∈ N , and R ∞ 0 . . . R ∞ 0   ϕ ( u 2 1 , . . . , u 2 n )   Q n k =1 u d k − 1 k du 1 . . . du n < + ∞ for some d k ∈ N , k = 1 , . . . , n . L et h k , b k ∈ C ( E k ) , h k stritly p ositive in their ar guments for al l k = 1 , . . . , n . Then K ( x 1 , . . . , x n , t 1 , . . . , t n ) := ϕ  k x 1 k 2 2 h 1 ( t 1 ) , . . . , k x n k 2 2 h n ( t n )  n Y k =1 b k ( t k ) ∈ Φ( R d 1 × . . . × R d n × E 1 × . . . × E n ) if, and only if, g d 1 ,...,d n  s 1 p h 1 ( t 1 ) , . . . , s n p h n ( t n )  n Y k =1 b k ( t k )  h k ( t k )  d k / 2 ∈ Φ( E 1 × . . . × E n ) for every s k ≥ 0 , k = 1 , . . . , n , wher e g d 1 ,...,d n ( s 1 , . . . , s n ) := Z ∞ 0 . . . Z ∞ 0 ϕ ( u 2 1 , . . . , u 2 n ) n Y k =1 u d k − 1 k j d k / 2 − 1 ( s k u k ) du 1 . . . du n . Pr o of. The statemen t an b e pro v ed in a similar w a y as Lemma 2. Let P n + b e the set all nite nonnegativ e Borel measures on [0 , + ∞ ) n , n ∈ N , and L n :=  ϕ ( u 1 , . . . , u n ) = Z ∞ 0 . . . Z ∞ 0 e − ( u 1 v 1 + ... + u n v n ) dµ ( v 1 , . . . , v n ) , µ ∈ P n +  . Ob viously , L 1 = C [0 , + ∞ ) T M (0 , + ∞ ) and Q n k =1 ϕ k ( u k ) ∈ L n for ev ery ϕ k ∈ L 1 , k = 1 , . . . , n . 11 Theorem 7. L et n ∈ N . F or al l k = 1 , . . . , n , let E k b e line ar sp a es, h k stritly p ositive funtions suh that h k ∈ C ( E k ) and d k ∈ N . Then, the fol lowing statements ar e e quivalent: 1. K ( x 1 , . . . , x n , t 1 , . . . , t n ) := ϕ  k x 1 k 2 2 h 1 ( t 1 ) , . . . , k x n k 2 2 h n ( t n )  Q n k =1  h k ( t k )  − d k / 2 ∈ Φ( R d 1 × . . . × R d n × E 1 × . . . × E n ) ∀ ϕ ∈ L n . 2. e − λh k ( t k ) ∈ Φ( E k ) , ∀ λ > 0 , k = 1 , . . . , n . Pr o of. Let us pro v e the impliation (1) = ⇒ (2) . F or ev ery xed k = 1 , . . . , n in ondition 1 , w e tak e ϕ ( u 1 , . . . , u n ) = ϕ k ( u k ) , ϕ k ∈ L 1 , and t i = 0 ∈ E i for i 6 = k . Then ϕ k  || x k || 2 2 h k ( t k )  ( h k ( t k )) − d k / 2 ∈ Φ( R d k × E k ) ∀ ϕ k ∈ L 1 . By Theorem 3 w e get e − λh k ( t k ) ∈ Φ( E k ) ∀ λ > 0 . Let us no w pro v e the rev erse impliation. Let e − λh k ( t k ) ∈ Φ( E k ) , ∀ λ > 0 , k = 1 , . . . , n . 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