Tapered Dirichlet and Fejér kernels

Transcription

1 Tapered Dirichlet and Fejér kernels David Casado de Lucas January 23, 2009 Contents Introduction 2 Definitions 2 3 Classical versions 2 4 Taper coefficients 3 4. Definition Properties The tapered Dirichlet kernel 6 5. Definition Properties The tapered Fejér kernel 2 6. Definition Properties Relation between the kernels 9 8 Summary of expressions 20 9 aïve applications 22 Introduction Tapered versions of the classical Dirichlet s and Fejér s kernels are defined. The expressions as trigonometric series make some properties trivial. For some others to be obtained, coefficients capturing the taper are used. The majority of the properties provided have a descriptive character. The most important, from a theoretical point of view, are: Item 6 of Lemma Item 7 of Lemma 3 Item 8 of Lemma 3

2 Item of Lemma 3 Item of Lemma 4 Item 3 of Lemma 4 2 Definitions Definition Dahlhaus, 997) For a complex-valued function f, H f ), λ) : s0 fs)e iλs, ) and, for the data taper hx), ) ) k H k, λ) : H h, λ 2) and H λ) H, λ). 3) Definition 2 Dahlhaus, 997) For the data taper hx), H k : 0 hu) k du. 4) Remark r ) k h H k,0) H k. 5) r0 3 Classical versions The classical Dirichlet s and Fejér s kernels are, for M, respectively, Definition 3 D M µ) : +M n M e iµn, 6) and Definition 4 F M µ) : M [D 0µ) + D µ) + + D M µ)] M 2 M m0 D m µ). 7)

3 Some other expressions of these kernels are [ ] D M µ) M + 2 cosµn), 8) n and D M µ) F M µ) [ F M µ) + 2 +M ) n M ) sin[m + /2)µ], 9) sin[µ/2] M n n ) e iµn, 0) M ] n ) cosµn), ) M F M µ) M ) 2 sin[mµ/2]. 2) sin[µ/2] 4 Taper coefficients In this section coefficients describing the taper are defined and some of its properties obtained. 4. Definition It can be written H k, µ) 2 H k, µ)h k, µ) + ) n ) e iµn r0 {r sn} s0 h ) r k h s ) ke iµr s) h ) r k h s ) k + ) n ) c n,k, e iµn 3) where Definition 5 c n,k, : {r sn} h r ) k h s ) k, 4) for n 0, +,,..., + ), ). 3

4 By applying again the same idea, H k, µ) 4 + ) n ) + ) c n,k,e iµn + ) n ) n 2 ) +2 ) n 2 ) + ) n 2 ) c n2,k,e iµn 2 c n,k,c n2,k,e iµn +n 2 ) d n,k, e iµn 5) where Definition 6 d n,k, : c n,k,c n2,k,, 6) {n +n 2 n} for n 0, +,,..., +2 ), 2 ). 4.2 Properties Lemma. From the symmetry of the expression, c n,k, c n,k, 2. By definition, c 0,k, H 2k, 0) 3. From 3) and H k, 0) R, H k, 0) 2 + ) n ) c n,k, 4. From 2, H k, 0) 2 c 0,k, + 2 n c n,k, 5. From 3) and 2, H k, µ) 2 c 0,k, + 2 c n,k, cosµn) n 4

5 6. For any n, n ) c n,k, n {r sn} h ) r k h s ) k n rn n rn h r ) k h r n ) k h r ) k h r n ) k 7) Thus, if h ) is continuous, n ) c n,k, H 2k 8) when From the orthogonality of the trigonometric system, so 8. The previous expression provides H k, µ) 2 e iµn dµ c n,k, c n,k, H k, µ) 2 e iµn dµ. 9) c n,k, c 0,k, 20) 9. When there is no taper c n,k, n 2) Lemma 2. From the symmetry of the expression, 2. From the definition and of Lemma, d 0,k, + ) n ) 3. From 5) and H k, 0) R, H k, 0) 4 d n,k, d n,k, c n,k, c n,k, +2 ) n 2 ) + ) n ) c 2 n,k,. d n,k, 22) 5

6 4. From the orthogonality of the trigonometric system, so H k, µ) 4 e iµn dµ d n,k, d n,k, H k, µ) 4 e iµn dµ. 23) Remark. The coefficients c n,k, previously defined are important to understand the behaviour of the tapered kernels that we define. It is easy to extract some qualitative information from the definitions. By hypothesis lim u 0 hu) 0 and lim u hu) 0; then, for and n big enough the terms h r ) h s ) in {r sn} h r ) k h s and {r sn} r s)l h r ) k h s ) k tend to zero. At the same time the exponent k contributes to accelerate the convergence. ) k 5 The tapered Dirichlet kernel 5. Definition For + and M 0, +,..., + ) the tapered Dirichlet kernel is defined as Definition 7 or L M,k, µ) : P r0 h r ) 2k L M,k, µ) : +M m M +M m M m ) {r sn} m ) P {r sn} h r ) k h s ) k h r ) k h s ) k e iµm 24) P r0 h r ) 2k e iµm 25) The name of this kernel is based on the fact that L M,k, µ) is equal to the classical Dirichlet kernel when there is no taper, that is, for hx) I{[0, ]} with I{} being the characteristic function. As in this case c n,k, n, then m ) c m,k, c 0,k, and the dependence on, the parameter of the taper, disappears: L M,k, µ) D M µ) 26) In fact {L M,k, µ)} M, can be seen as a triangular array of functions; with the formal assumption that h0) 0, the first element is L 0,k, µ). The parameter is related with the taper, while M is related with the kernel structure itself. Asymptotically it is not a strong restriction for to be bigger than M. When there is no taper the triangular array {L M,k, µ)} M, collapses into the classical sequence {D M µ)} M. In terms of the taper coefficients, this kernel can be written as L M,k, µ) +M m M m ) c m,k, c 0,k, e iµm 27) 6

7 or [ L M,k, µ) + 2 M m m) c m,k, c 0,k, cosµm) ]. 28) 5.2 Properties Lemma 3. L M,k, µ) is continuous on µ 2. L M,k, µ) is even, since L M,k, µ) L M,k, µ) 3. L M,k, µ) R, since L M,k, µ) L M,k, µ) L M,k, µ) 4. L M,k, µ) is -periodic, since so the exponential function is 5. lim µ 0 L M,k, µ) +M m M continuous on µ m ) c m,k, c 0,k, L M,k, 0), since L M,k, µ) is 6. For any M, from 8) it holds that lim + L M,k, 0) +M m M 2M + ) 7. More generally, for any M, from 27) and 8), it holds that lim + L M,k, µ) +M m M e iµm D M µ). That is, L M,k, µ) is asymptotically as close to D M as desired. 8. From the previous property, lim M + L M,k, µ) lim M + D M µ) δ0) notice that by definition it is necessary for to be bigger than M) 9. It holds that L M,k, µ) L M,k, 0) when i) h ) is non-negative or non-positive, or ii) k is even. In these cases sup µ L M,k, µ) L M,k, 0) 7

8 0. From 23 below, L M,k, µ) +M m M L M,k, γ)e iγm dγe iµm L M,k, γ) +M m M eiγ µ)m dγ L M,k, γ)d M µ γ)dγ L M,k, γ)d M µ γ)dγ D M L M,k, )µ). 29) Since in these steps the continuity or derivability of h ) have not been used, as a particular case D M µ) D M D M )µ).. From, 6 and 8 below it seems that L M,k, µ) δ0) when +, where δ ) is the Dirac s delta function. To prove this important result, item 8 could be called, but a direct argument is the following: L M,k, µ) +M n ) c n,k, e iµn c 0,k, n M + n e iµn δ0) when M + notice that, by definition, is bigger than M). From 8), n ) c n,k, c 0,k, 2. L M,k, µ) is derivable and L l) M,k, µ) P r0 h r ) 2k +M m M m ) {r sm} h r ) k h s ) k im) l e iµm P r0 h r ) 2k +M m M im) l m ) {r sm} h r ) k h s ) k e iµm c 0,k, +M m M im) l m ) c m,k,e iµm +M m M im) l m ) c m,k, e iµm 30) c 0,k, L l) M,k, is a continuous, even and -periodic function on µ. Besides, Ll) M,k, µ) R since from the symmetry of the expression L l) M,k, µ) Ll) M,k, µ). 8

9 On the other hand, L M,k, µ) is an analytic function, as it is a finite sum of analytic functions 3. As a rough bound, when i) h ) is non-negative or non-positive, or ii) k is even, L l) M,k, µ) M l L M,k, 0) 4. For l odd, L l) M,k, 0) 0 from the symmetry of the expression 5. For l even, when i) h ) is non-negative or non-positive, or ii) k is even, it holds that L l) M,k, 0) 0 if il < 0, and L l) M,k, 0) 0 if il > 0 6. With usual taper functions L M,k, 0) 0; that is, L M,k,µ) is not lineal in µ 0 or, geometrically, the curvature of the graph is not null in µ 0. With this taper functions L M,k, 0) < 0 and L M,k,µ) has a maximum in µ 0 when i) h ) is non-negative or non-positive or ii) k is even see the previous property) 7. With 30) and some computes, L l) M,k, µ) D M L l) M,k, )µ) 3) 8. It holds that L M,k, µ)dµ. Expression 24) highlights which the normalization factor is. 9. From 2 and 8, 0 L M,k, µ)dµ 2 L M,k, µ)dµ 20. Incomplete) Let {a } be a sequence of positive real numbers such that a 0; since L M,k, µ)dµ L M,k, µ)dµ + µ a L M,k, µ)dµ µ >a when the second integral tends to zero, as a 0, for any a > 0 there exists 0 such that a a when > 0, and in this case L M,k, µ)dµ L M,k, µ)dµ 0. µ >a µ >a Then, to prove the asymptotic negligibility of the tails it is enough to find a sequence {a } such that the first integral of the right hand tends to. Since odd derivatives of L M,k, in µ 0 are null, L M,k, µ) L M,k, 0) + 2 L M,k,µ 0 )µ 2 with µ 0 0, µ) 9 0

10 then L M,k, µ)dµ 2a L M,k, 0) + µ a 2 L M,k,µ 0 ) µ 2 dµ µ a 2a L M,k, 0) + 3 L M,k,µ 0 )a 3. If a 2L M,k, 0)), 3 L M,k,0)a 3 C L L M,k, 0) 3 M,k, 0) so it seems it is necessary a better bound than 3 for L M,k, 0) As for the classical Dirichlet s kernel, the tapered Lebesgue constants could be defined and studied: L M,k, 22. L M,k, L p [, +π], although perhaps L M,k, p + with M 23. From 27), c n,k, n ) c 0,k, L M,k, γ)e iγn dγ n ) c 0,k, L M,k, γ)e iγn dγ 32) 24. Also from 27), L M,k, µ) 2 dµ +M m M m ) c m,k, c 0,k, ) From 30), L l) M,k, µ)dµ 0. 33) This expresses the symmetry of L M,k, 26. As L M,k, L [, +π], the Fourier coefficients can be considered; for m Z, ˆL M,k, m) : L M,k, µ)e iµm dµ m ) c m,k, c 0,k, if m 0 if m > 34) Then, 27) L M,k, µ) +M) m M ˆL M,k, m)e iµm m Z ˆL M,k, m)e iµm 35) 0

11 and 24) L M,k, µ) 2 dµ +M m M) ˆL M,k, m) 2 m Z ˆL M,k, m) 2 36) That is, the Fourier series expression and the Parseval s equality had been found. otice that ˆL M,k, m) L M,k, µ)dµ, 37) so from 34) c m,k, m ) c 0,k, m Z. On the other hand, the Fourier partial sums, in general, are S M f)µ) : +M m M ˆfm)e iµm fγ)d M µ γ)dγ 38) which for the tapered Dirichlet kernel, with M, provides S M L M,k, )µ) as steps 29) states For the derivatives, directly from 35), +M L M,k, γ)d M µ γ)dγ L M,k, µ) 39) L l) M,k, µ) m M im) l ˆLM,k, m)e iµm im) l ˆLM,k, m)e iµm 40) m Z or, from 30) and 34), L l) M,k, µ) +M m M +M m M i) l n l m ) c n,k, c 0,k, e iµm in) l ˆLM,k, n)e iµm. 4) To see directly that L M,k, µ) is an analytic function there is another justification in 2) for µ close to µ 0 L M,k, µ) +M m M ˆL M,k, m)e iµm +M m M +M m M l0 l! ˆL M,k, m)e iµ 0m e iµ µ 0)m ˆL M,k, m)e iµ 0m +M m M im) l µ µ 0 ) l l! l0 im) l ˆLM,k, m)e iµ 0m µ µ 0 ) l L l) M,k, µ 0) µ µ 0 ) l l! l0

12 6 The tapered Fejér kernel 6. Definition The tapered Fejér kernel is defined, for +, as Definition 8 or K k, µ) : K k, µ) : P h r0 r ) 2k r0 s0 H 2k, 0) H k,µ)h k, µ) h ) r k h s ) ke iµr s) 42) H 2k, 0) H k,µ) 2. 43) The name of the kernel is based on the fact that K k, µ) is equal to the classical Fejér kernel when there is no taper, that is, when hx) I{[0, ]} with I{} being the characteristic function. Since in the double sum r0 s0 there are terms with r s 0, terms with r s and with r s, 2 terms with r s 2 and 2 with r s 2,..., and terms with r s + ) and with r s ), it can be written r0 s0 e iµr s) + ) n ) e iµn + + 2) n 2) e iµn n e iµn + D µ) + D 2 µ) + + D µ) + D 0 µ) F µ), 44) where D ) and F ) are, respectively, the classical Dirichlet s and Fejér s kernels. Then, in this particular case K k, µ) F µ) F µ). In terms of the taper coefficients, this kernel can be written as K k, µ) c 0,k, + ) n ) c n,k, e iµn or K k, µ) + ) n ) + ) n ) c n,k, c 0,k, e iµn 45) n ) c n,k, c 0,k, [ c 0,k, + 2 H 2k, 0) [ [ n n n c n,k, c 0,k, cosµn) n) c n,k, n ) e iµn 46) c n,k, cosµn) ] c 0,k, n ] ] ) cosµn). 47) 2

13 6.2 Properties Lemma 4. K k, µ) is continuous on µ 2. K k, µ) is even, since from the symmetry of its expression K µ) K µ) 3. K k, µ) R by definition. Besides, K k, µ) K k, µ) K k, µ) 4. K k, µ) 0 by definition 5. K k, µ) is -periodic, since so the exponential function is 6. lim µ 0 K k, µ) H H 2k, 0) k,0)h k, 0) H k, 0) 2 K H 2k, 0) k,0), since H k, µ) is continuous on µ 7. As H k, 0) H k when +, K k, 0) H k,0) 2 H 2k, 0) 2 H k, 0) 2 H 2k, 0) + when +. Besides, K k, 0) for large values of. 8. It holds that K k, µ) K k, 0) when i) h ) is non-negative or non-positive, or ii) k is even, since H k, µ) H k, 0). In these cases sup µ K k, µ) K k, 0) 9. From the Cauchy-Schwarz s inequality, when i) h ) is non-negative or nonpositive, or ii) k is even, K k, µ) h 2 ) r ke iµr H 2k, 0) H 2k, 0) H 2k,0). r0 This result can also be obtained from 8 and 7, since the mentioned inequality implies under i) or ii) that H k, 0) 2 H 2k, 0). On the other hand H 2 k H 2k by Jensen s inequality.) 0. From the definition and 5), K k, µ) 2 [H 2k, 0)] 2 ) 2 +2 ) n 2 ) +2 ) n 2 ) d n,k, e iµn d n,k, e iµn. 48) c 2 0,k, 3

14 . From 46) and 8), K k, is asymptotically as close to F as desired. In fact, both of them tend to the Dirac s delta function. 2. From 7 of Lemma K k, µ) + ) n ) + ) n ) c n,k, c 0,k, e iµn K k, γ)e iγn dγe iµn K k, γ) + ) n ) eiγ µ)n dγ K k, γ)d µ γ)dγ K k, γ)d µ γ)dγ D K k, )µ). 49) Since in these steps the continuity or derivability of h ) have not been used, as a particular case F µ) D F )µ). 3. From, 7 and 20 below it seems that K k, µ) δ0) when +, where δ ) is the Dirac s delta function. To prove this important result, item could be called, but a direct argument is the following: K k, µ) + ) n ) + n n ) c n,k, c 0,k, e iµn δ0) n ) e iµn when +. On the one hand, n ) c n,k, c 0,k, see 6 of Lemma ); on the other hand, n / for any n a similar argument is used, for example, by Priestley [98] in Theorem 4.8. and in section ) 4. K k, µ) is derivable and K l) k, µ) H 2k, 0) + ) n ) r0 s0 i) l n l c n,k, [ ir s)] l h r ) k h s ) ke iµr s) c 0,k, e iµn. 50) 4

15 K l) k, is a continuous, even and -periodic function on µ. Besides, Kl) k, µ) R since from the symmetry of the expression K l) k, µ) Kl) k, µ). On the other hand, K k, µ) is an analytic function, as it is a finite sum of analytic functions 5. As a rough bound, when i) h ) is non-negative or non-positive, or ii) k is even, K l) k, µ) )l K k, 0) < l K k, 0) 6. For l odd, K l) k, 0) 0 from the symmetry of the expression 7. For l even, when i) h ) is non-negative or non-positive, or ii) k is even, it holds that K l) k, 0) 0 if il < 0, and K l) k, 0) 0 if il > 0 8. With usual taper functions K k, 0) 0; that is, K k,µ) is not lineal in µ 0 or, geometrically, the curvature of the graph is not null in µ 0. With this taper functions K k, 0) < 0 and K k,µ) has a maximum in µ 0 when i) h ) is non-negative or non-positive or ii) k is even see the previous property) 9. With 50) and some computes, 20. It holds that K l) k, µ) D K l) k, )µ) 5) K k, µ)dµ H 2k, 0) r0 s0 h ) r 2k H 2k, 0) r0 h ) r k h s ) +π k e iµr s) dµ Expression 42) highlights which the normalization factor is. 2. From 2 and 20, 0 K k, µ)dµ 2 K k, µ)dµ Incomplete) Let {a } be a sequence of positive real numbers such that a 0; since K k, µ)dµ K k, µ)dµ + µ a K k, µ)dµ µ >a when the second integral tends to zero, as a 0, for any a > 0 there exists 0 such that a a when > 0, and in this case K k, µ)dµ µ >a K k, µ)dµ 0. µ >a 5

16 Then, to prove the asymptotic negligibility of the tails it is enough to find a sequence {a } such that the first integral of the right hand tends to. Since odd derivatives of K k, in µ 0 are null, K k, µ) K k, 0) + 2 K k,µ 0 )µ 2 with µ 0 0, µ) then K k, µ)dµ 2a K k, 0) + µ a 2 K k,µ 0 ) µ 2 dµ µ a 2a K k, 0) + 3 K k,µ 0 )a 3. If a 2K k, 0)), 3 K k,0)a 3 C K K k, 0) 3 k, 0) so it seems it is necessary a better bound than 5 for K k, 0) K k, L p [, +π], although perhaps K k, p + with 24. From 45), or directly from 7 of Lemma, c n,k, c 0,k, K k, γ)e iγn dγ 52) 25. From 48), K k, µ) 2 dµ +π ) 2 d 0,k, ) 2 c 2 0,k, +2 ) n 2 ) d n,k, e iµn dµ c 2 0,k, + ) n ) cn,k, This result can also be obtained directly from 45) c 0,k, ) 2. 53) 26. From 50), This expresses the symmetry of K k,. K l) k, µ)dµ 0. 54) 27. Incomplete) It holds that, for j and l fixed, K k, µ)e iµt j t l ) dµ c 0,k, + ) n ) c n,k,e iµn e iµt j t l ) dµ c tj t c l,k, c t j t l,k, I{ t j t l < }, 0,k, c 0,k, 6

17 where I{} is the characteristic function. That is, only close points provide a value different from zero for the previous integral. In this item the notation t j, t l, S,... is the used in Casado????). ow, for j fixed it can be defined the influence interval of point t j as I j : {t l / t j t l Sj l) < } 55) and it holds that {t l / j l) < } {t l / t l t j } if S I j {t l / Sj l) < } {t l / j l < /S} if S/ 0 In I j there are only one point when S and 2/S points when S/ 0 notice that in this case /S ). There is a subtle difference between I j defined in 55) and B j defined in Casado????): the length is given by definition for the former and imposed by orthogonality properties of the trigonometric functions in the latter. ow, the sum on l is M l K k, µ)e iµt j t l ) dµ M l c tj t l,k, c 0,k, I{t l I j }. c 0,k, c 0,k, if S P M l c t j t l,k,i{t l I j } c 0,k, if S/ 0 56) if S... if S/ 0 57) 28. As K k, L [, +π], the Fourier coefficients can be considered; for m Z, c ˆK k, m) : m,k, +π K k, µ)e iµm c 0,k, if m dµ 58) 0 if m > Then, 45) K k, µ) + ) m ) ˆK k, m)e iµm m Z ˆK k, m)e iµm 59) and 53) K k, µ) 2 dµ + ) m ) ˆK k, m) 2 m Z ˆK k, m) 2 60) That is, the Fourier series expression and the Parseval s equality had been found. otice that ˆK k, m) K k, µ)dµ, 6) 7

18 so from 58) c m,k, c 0,k, m Z This result was also obtained directy from 9). On the other hand, the Fourier partial sums, in general, are S M f)µ) : +M m M ˆfm)e iµm fγ)d M µ γ)dγ 62) which for the tapered Fejér kernel, with M, provides S K k, )µ) as steps 49) states. For the derivatives, directly from 59), K l) k, µ) + ) or, from 50) and 58), m ) im) l ˆKk, m)e iµm m Z K l) k, µ) K k, γ)d µ γ)dγ K k, µ) 63) + ) n ) i) l n l c n,k, c 0,k, im) l ˆKk, m)e iµm 64) e iµn + ) n ) in) l ˆKk, n)e iµn. 65) To see directly that K k, µ) is an analytic function there is another justification in 4), for µ close to µ 0 K k, µ) + ) m ) + ) m ) ˆK k, m)e iµm ˆK k, m)e iµ 0m e iµ µ 0)m + ) m ) ˆK k, m)e iµ 0m im) l µ µ 0 ) l l! l0 l0 l0 l! K l) + ) m ) k, µ 0) l! im) l ˆKk, m)e iµ 0m µ µ 0 ) l µ µ 0 ) l 8

19 7 Relation between the kernels Since {L M,k, µ)} M, is a triangular array of functions, sums in several directions can be studied. The results of these sums have been represented in Figure ). An important general relation between the tapered Dirichlet and Fejér kernels is that the latter can be expressed as a Cesàro-like sum from the former. For the -th row, with the same idea used to obtain 44), or using that for a general sequence of numbers +i i0 j i a j + ) j ) j )a j, it can be written L m,k, µ) m0 M0 M0 m M +M m ) c m,k, e iµm c 0,k, m M +M + ) n ) + ) n ) m ) c m,k, c 0,k, e iµm n ) n ) c n,k, c n,k, c 0,k, e iµn c 0,k, e iµn K k, µ) 66) The previous average is not the classical Cesàro sumation, as the terms L M,k, µ) depends on. The sum on M explains why the kernel K k, does not depend explicitly on M, only on the taper parameter ; nevertheless, there is an implicit relation between M and in the definition of the triangular array. By using the Fejér sums notation, the previous result can be expressed as σ L M,k, K k,. 67) The sum also shows that the first element of the sequence {K k, } is K k, with the formal assumption that h0) 0, as it was done for L M,k, ). Finally, results 66) and of Lemma 3 are a new proof of 3 of Lemma 4. On the other direction, if the sum is applied to the l-th column, + l L M,k,n µ) nl + l +M m M nl + l +M n m ) c m,k,n e iµm n c 0,k,n m M nl n m ) c m,k,n e iµm n c 0,k,n otice that, for the coefficients, + l nl +M m M e iµm D M µ) 68) n m ) c m,k,n n c 0,k,n 9

20 Figure : Triangular arrays of kernels, with and without taper, and the averages or its limits) in the margin since, from 8), n m ) c m,k,n n c 0,k,n. Another way to justify this is to consider the item 7 of Lemma 3 and the properties of the Cesàro sumation. As expected, when there is no taper these coefficients take the value and the limit becomes an equality. Finally, only the diagonal of the triangular array can be taken into account this can be seen as a sort of synchronization of the parameters). Thus, for the sequence {L M,k,M+ µ)} M {L,k, µ)}, from 8), L,k, µ) + ) m ) m ) c m,k, c 0,k, e iµm + m e iµm δ. 69) A different kind of relation can be obtained from the integral expressions 32) and 52); for any n n ) L M,k, γ)e iγn dγ K k, γ)e iγn dγ 70) 8 Summary of expressions At this point it is interested to summarize some expressions: 20

21 Classical Dirichlet s kernel D M µ) : [ D M µ) + 2 +M n M e iµn [6] ] M cosµn) n [8] Classical Fejér s kernel F M µ) F M µ) : M [ F M µ) + 2 +M ) n M ) M n M m0 D m µ) [7] n ) e iµn [0] M ] n ) cosµn) M [] Tapered Dirichlet kernel L M,k, µ) +M n M [ L M,k, µ) + 2 M n n ) c n,k, c 0,k, e iµn [27] n) c n,k, c 0,k, cosµn) ] [28] Tapered Fejér kernel K k, µ) K k, µ) + ) n ) m0 n ) c n,k, c 0,k, L m,k, µ) [66] n ) e iµn [46] [ ] K k, µ) n) + 2 c n,k, n ) cosµn) c n 0,k, [47] That is, the tapered kernels can be expressed in the same type of sums and fulfill the similar relations than the classical ones: 2

22 Compare expressions 27) and 28) with 6) and 8). Compare also expressions 46) and 47) with 0) and ), respectively. Compare expression 66) with 7). Finally, compare the relation between expressions 27) and 28) with 46) and 47), respectively, with that between expressions 6) and 8) with 0) and ). 9 aïve applications Both the tapered Dirichlet and Fejér kernels are symmetrical and tend to the Dirac s del function when M. For such kernels. The following point convergence holds when M + 0 when µ 0 fµ)s M µ) 0 when µ 0 and fµ) 0 when µ 0 and fµ) 0 2. The following weak or in distribution) convergence holds when M + b a S Mµ)dµ b a fµ)s Mµ)dµ if a < 0 < b 2 if a 0 or b 0 0 if b < 0 or a > 0 3. The following weak or in distribution) convergence holds when M + f0) if a < 0 < b 4. Since b a fµ)s M µ)dµ 2 f0) if a 0 or b 0 0 if b < 0 or a > 0 a fµ)s M µ)dµ fµ)s M µ)dµ fµ)s M µ)dµ, b the asymptotic weak or in distribution) convergence, when M +, of this relation can be described symbolically as f0) f0) 0 0 if a < 0 < b f0) f0) f0) 0 if a f0) f0) 0 f0) if b f0) 0 f0) if b < 0 0 f0) f0) 0 if a > 0 22

23 5. If f exists b a when M +. fµ)s M µ)dµ b a fµ) S M µ)dµ b fµ) S M µ)dµ a fµ) if a < 0 < b 2 fµ) if a 0 or b 0 0 if b < 0 or a > 0 6. The previous tapered kernels can allow for some existing proofs to be improved in the taper framework. This could provide slightly but crucial better rates of convergence. In Dahlhaus 997) one finds an example: We conjecture that the rate O 2 ) cannot be improved with a periodogram type estimator. A periodogram without taper would lead to a bias of O ) and therefore to / 0 which contradicts / 0. Thus, without taper it is not possible to achieve T -consistency at all. It is noteworthy that the use of a data taper does not lead to an increase of the variance if S/ 0. The improvement of such existing proofs would be based on the idea of introducing the taper in the process and changing asymptotically, in the proof, the tapered versions of the kernels by the classical ones wherever the former appear. 7. Another application takes place in the proof that has motivated these definitions in Casado,????) References [] Casado, D.????). Asymptotic estimation for the non-stationary integrated spectrum [2] Dahlhaus, R. 997). Fitting time series models to nonstationary processes The Annals of Statistics. 25 ), 37. [3] Priestley, M.B. 98). Spectral Analysis and Time Series. Elsevier Eleventh printing 200. Reprinted 2004). 23