Mathematical Notes, vol. 74, no. 2, 2003, pp. 232–244. Translated from Matematicheskie Zametki, vol. 74, no. 2, 2003, pp. 242–256. c Original Russian Text Copyright 2003 by K. I. Oskolkov.
On a Result of Telyakovskii and Multiple Hilbert Transforms with Polynomial Phases K. I. Oskolkov Received January 29, 2003
Abstract—In this paper, we prove a multiple analog of the theorem proved by Arkhipov and the author in 1987, which provides an estimate for the discrete Hilbert transform with polynomial phase. For the linear case, the corresponding estimates of the sum of multiple trigonometric series was proved by Telyakovskii. Key words: Fourier series, discrete Hilbert transform, multiple trigonometric series, polyno-
mial phase.
I dedicate this paper to my teacher Sergei Alexandrovich Telyakovskii on the occasion of his seventieth birthday.
Suppose that Rd , d = 1, 2, . . . , is the d-dimensional real Euclidean space of vectors v = (v1 , . . . , vd ) ; Zd is the lattice of integers in Rd ; Nd is the subset of vectors n ∈ Rd with integer coordinates nk ; Zd /N is the set of vectors in Rd with rational coordinates. We say that a subset ω ⊂ Rd is coordinate-wise convex, if the intersection of ω with any line which is parallel to one of the coordinate axes is either an interval or the empty set. Denote by Ωd the class of all coordinate-wise convex sets in Rd . Telyakovskii [1, 2] proved the following remarkable result on multiple sine-series with linear phase. Theorem 1. For d ≥ 2 the following estimate holds: sup sup d d
ω∈Ω
x∈R
n∈ω∩Nd
sin nd xd sin n1 x1 ... < ∞. n1 nd
This theorem has numerous applications. In particular, it allows us to estimate the Kolmogorov width of classes of functions with bounded mixed derivative and, in general, it is applied to hyperbolic cross approximations as well as in the convergence theory of multiple Fourier series of functions of several variables with variation bounded in the Hardy sense (see [3]). The following result was proved in [4] by using the Vinogradov method of trigonometric sums [5]. It is related to the one-dimensional discrete Hilbert transform with polynomial phase of the highest order (see also [6]). The symbol P r , r ∈ N , denotes the set of algebraic polynomials p of degree r in one variable with real coefficients, p(0) = 0 . 232
0001-4346/2003/7412-0232$25.00
c 2003 Plenum Publishing Corporation
ON A RESULT OF TELYAKOVSKII AND MULTIPLE HILBERT TRANSFORMS
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Theorem 2. For a polynomial p ∈ P r , we set hm (p) :=
m eip(n) − eip(−n) , n n=1
m ≥ 1.
Then sup sup |hm (p)| < ∞
(1)
p∈P r m
and for each fixed polynomial p ∈ P r , the following limit exists: h(p) := lim hm (p). m→∞
This result also has various applications, including solutions of problems on uniform convergence spectra and the analysis of properties of solutions to the Cauchy problem for a wide class of linear partial differential equations of Schr¨ odinger type with periodic initial conditions. Actually, Theorem 2 is the starting point for considering the wide class of Vinogradov series V (f ; x1 , . . . , xr ) :=
2πi(x1 n+x2 n2 +···+xr nr )
cn (f )e
,
cn (f ) :=
n∈Z
1
f (x)e−2πinx dx.
0
The reader can find a survey of results in these fields in the paper [7] (see [8, 9] for the latest developments). Here we will prove Theorem 3 on the multiple discrete Hilbert transform with polynomial phases; it generalizes and somewhat sharpens Theorems 1 and 2. We say that the number sequence f = {fn }n∈N is slow, if it satisfies the Littlewood–Paley condition (see [10, Chap. 15]):
f S := f ∞ + sup n
|fm − fm+1 | < ∞,
f ∞ := sup |fn |. n
n≤m≤2n
Denote by S the class of all sequences such that f S < ∞ . It is clear that if the sequence f is of bounded variation in the usual sense, i.e., f ∈ BV , then this sequence is slow. However, the class S is substantially wider than BV . For example, it is obvious that for any fixed real t = 0 , / BV . we have ft := {nit } = {eit ln n }n∈N ∈ S , but ft ∈ We say that the sequence f = {fn }n∈Nd (indexed by d -tuples of natural numbers) is coordinatewise slow (the corresponding notation is f ∈ S d ) if the restrictions of f to the lines parallel to the coordinate axes are uniformly slow:
f S d := f ∞ + max sup sup 1≤k≤d nk
n
|fnk +mek − fnk +(m+1)ek | < ∞,
n≤m≤2n
k
where ek := (0, . . . , 0, 1, 0, . . . , 0) , k = 1, . . . , d , denotes the standard basis in Rd , and for n = (n1 , . . . , nk , . . . , nd ) we use the notation nk := n − nk ek . It is obvious that the characteristic function of a coordinate-wise convex domain ω , i.e., ω ∈ Ωd is coordinate-wise slow. Further on, we also use the following notation: P r ,d is the set of collections of polynomials p = (p1 , . . . , pd ) , where pk ∈ P r ; m is the parallelepiped {n ∈ Nd : nk ≤ mk , k = 1, . . . , d} . r
r ,d
r
r ,d
The notation and in relations of the form A B and A B , respectively, indicates that the finite multipliers cr , cr ,d depending only on the given parameters ( r or r and d) exist, and for them we have |A| ≤ cr |B| or |A| ≤ cr ,d |B| . MATHEMATICAL NOTES
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K. I. OSKOLKOV
Theorem 3. Suppose that for p ∈ P r ,d and m ∈ Nd ,
hm (f , p ) :=
fn
n∈m
d eipk (nk ) − eipk (−nk ) . nk
k=1
Then f ∈ S d implies that r ,d
sup sup |hm (f , p )| f S d
m∈Nd
(2)
p
and for any collection of polynomials p ∈ P r ,d the limit h(f , p ) := limmink mk →∞ hm (f , p ) exists. It is interesting to compare this theorem with the following negative result proved recently by Graev in [11]: there exists a number x such that the following sequence diverges as N → ∞: hN :=
N N sin(mnx) . mn m=1 n=1
Proof of Theorem 3. For p ∈ P r , we set ip(t)
e−
:=
eip(t) − eip(−t) , 2i
p ∈ P r ,d consider the following trigonometric sums and integrals: and for n ∈ N , n ∈ Nd , Tn (p) :=
n 1 ip(n) 1 n ip(t) e− , In (p) := e dt, T0 (p) = I0 (p) := 0, An (p) := Tn (p) − Tn−1 (p), n m=1 n 0 − Bn (p) := Bn (p) − Bn−1 (p),
An ( p ) :=
d
Ank (pk ),
Bn ( p ) :=
k=1
We have
ip(n)
e− n
= An (p) +
Tn−1 (p) , n
n−1
Bnk (pk ).
k=1
ip(t)
n
d
e− t
In−1 (p) . n
(3)
|In−1 (p)| < ∞. n
(4)
dt = Bn (p) +
As proved in [4], (a)
sup p∈P r
|Tn−1 (p)| < ∞, n
(b)
sup p∈P r
n∈N
n∈N
Assertion (b) is related to Hilbert integral transforms g(p) := lim gm (p), m→∞
gm (p) :=
0
m
ip(t)
e− t
dt.
The uniform boundedness of supm>0 supp∈P r |gm (p)| < ∞ and the existence of the limit g(p) were proved by Stein and Wainger [12]. Assertion (a) in (4) is more difficult, because the arithmetical characteristics of the coefficients vector of the polynomial p play an important role and the proof requires the use of the Hardy– Littlewood–Vinogradov disk method. This assertion was proved in [4]. The global boundedness sup sup |hm (p)| < ∞
m>0 p∈P r
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(see Eq. (1)) was proved independently by Stein and Wainger a little bit later than in [4] (see [6] and [13, p. 373]). For a subset of indices A ⊂ [1, d] (possibly empty), we consider the compound integral-discrete products Ank (pk ) Bnk (pk ) , n ∈ Nd , p ∈ P r ,d . Cn (A, p ) := k∈A
k∈[1,d]\A
Then Eqs. (3) and (4) imply that Theorem 3 is a corollary corresponding to the special case A = [1, d] of the following assertion. Lemma 1. Suppose that f := {fn }n∈Nd is a coordinate-wise slow sequence, f S d ≤ 1 , while A ⊂ [1, d] is a subset of indices and p ∈ P r ,d . Then the limit S(f , A, p ) :=
lim
mink mk →∞
exists and
fn Cn (A, p )
n∈m
r ,d
|S(f , A, p )| 1.
(5)
Proof. For a polynomial p ∈ P r , we denote by x = (x1 , . . . , xr ) the coefficients vector of the polynomial p/(2π) and set p(t) = 2πP (x, t) := 2π(x1 t + · · · + xr tr ),
p∗ (t) := 2π(|x1 |t + · · · + |xr |tr ).
The following estimates are trivial: max(|Tn (p)|, |In (p)|) ≤ min(1, p∗ (n)),
|Cn (p)| ≤ max(|An (p)|, |Bn (p)|)
min(1, p∗ (n)) , n (6)
because |eip − | = | sin p− | , where p− (t) := (p(t) − p(−t))/2 and |p− | ≤ p∗ . We divide the proof of Lemma 1 into the four cases: (1) A = ∅ , d = 1 ; (2) A = ∅ , d ≥ 2 ; (3) A = {1} , d = 1 ; (4) a general set A ⊂ [1, d] , d ≥ 2 . Case (1) is the simplest case. Here one may just consider the sum of integrals S(f , p) = S(f , ∅, p) :=
fn Bn (p).
n∈N
By applying the Abel transform for the mth partial sum, we obtain Sm (f , p) :=
m
fn Bn (p) =
n=1
m
fn (In (p) − In−1 (p)) = fm Im (p) +
n=1
m−1
In (p)∆fn ,
n=1
∆fn := fn − fn+1 . Hence it suffices to prove that In (p) → 0 as n → ∞ and that f ∈ S , f S ≤ 1 , implies ∞
r
|In (p)∆fn | 1.
n=1
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K. I. OSKOLKOV
The definition of slow convergence implies that m
p∗ (n)|∆fn | p∗ (m),
n=1
∞
ρ
−ρ p−ρ ∗ (n)|∆fn | p∗ (m),
p ∈ Pr ,
ρ > 0.
(8)
n=m+1
Next, for the integral In , the following estimate holds (see also [4, 8]): |In (p)| min p∗ (n), p−ρ ∗ (n) ,
ρ :=
1 . r
(9)
Because of Eq. (6), this estimate is a consequence of the inequality n 1 ip(t) e dt min 1, p−ρ p ∈ Pr , ∗ (n) , n 0 which is equivalent to the well known Vinogradov estimate of a standard oscillatory integral with polynomial phase (see [5, Chap. 2, Lemma 4]):
1
ip(t)
e 0
dt =
1
2πiP (x, t)
e
0
dt ≤ min 1, 32 min |xk |−ρ ≤ min 1, 32r ρ p−ρ ∗ (1) . k
Hence inequality (9) implies that, indeed, In (p) → 0 as n → ∞ , and ∞
|In (p)∆fn |
n=1
∞
r r −ρ min p∗ (n), p−ρ ∗ (n) |∆fn | min p∗ (m) + p∗ (m) 1. m
n=1
Moreover, we have ∞ ∞ fn Bn (p) ≤ |fm Im (p)| + |In (p)∆fn | |S(f , p) − Sm (f , p)| = n=m
∞
n=m
r −ρ min p∗ (n), p−ρ ∗ (n) |∆fn | min(1, p∗ (m)),
(10)
n=m
which gives an estimate of the convergence rate of the integral in case (1) and concludes the proof of Lemma 1 in case (1). Case (2). Now we must consider the multiple sum of integrals S(f , p ) = S(f , ∅, p ) =
fn Bn ( p ).
n∈Nd
We will apply a modification of Telyakovskii’s key idea used in the proof of Theorem 1. For a given collection of polynomials p = (p1 , . . . , pd ) ∈ P r ,d , we split the summation domain Nd into d! subdomains corresponding to the values of the polynomials p∗1 , . . . , p∗d . Typical subdomains are the “algebraic octants” ω = ω( p ) := {(n1 , . . . , nd ) ∈ Nd : p∗1 (n1 ) ≥ · · · ≥ p∗d (nd )}, p ) := {(n1 , . . . , nd ) ∈ Nd : p∗1 (n1 ) ≥ · · · ≥ p∗d (nd ), nd ≥ m}, ωm = ωm (
m ∈ N.
(11)
Other octants (whose total number equals d! − 1) are generated by ω by various permutations of inequalities between the polynomials p∗ k (nk ) and by replacement of the inequalities ≤ by strict MATHEMATICAL NOTES
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inequalities < in all or in a part of relations (11), so that the subsets obtained in this way have no common points and split the total set Nd . Let us prove that for a compactly supported sequence f such that f S d ≤ 1 , the following estimate holds: r ,d m ∈ N, Sωm := fn Bn ( p ). (12) |Sωm | min 1, p∗ −ρ d (m) , n∈ωm
For brevity, we simply write n1 := n , p1 := p and for n = (n, n2 , . . . , nd ) ∈ Nd , m ∈ N , k = 1, . . . , d − 1 we set nk := (nk+1 , . . . , nd ) ∈ Nd−k , d
p ) := Bnk (
d min(1, p∗ l (nl )) Πnk ( p ) := , nl
Bnl (pl ),
l=k+1 k ωm
:= {n ∈ N k
Then S ωm =
l=k+1
d−k
: p∗ k+1 (nk+1 ) ≥ · · · ≥ p∗ d (nd ), nd ≥ m}.
(13)
Bn1 ( p)
(14)
1 n1 ∈ωm
fn Bn (p) ,
n : p∗ (n)≥p∗ 2 (n2 )
and by Eqs. (10) and (8), we have r min 1, p∗ −ρ (n2 ) , f B (p) n n 2
d
|Bn1 ( p )| Πn1 ( p ).
n : p∗ (n)≥p∗ 2 (n2 )
Hence relation (12) follows from the chain of inequalities: r ,d
d | |Sωm
1 n1 ∈ωm
=
2 n2 ∈ωm
r
Πn1 ( p ) min 1, p∗ −ρ 2 (n2 )
Πn2 ( p)
n2 : p∗2 (n2 )≥p∗3 (n3 )
min p∗2 (n2 ), p∗ −ρ 2 (n2 ) n2
r ,d Πn2 ( p ) min 1, p∗ −ρ 3 (n3 ) · · ·
2 n2 ∈ωm
∞ r min p∗ d (nd ), p∗ −ρ d (nd ) min 1, p∗ −ρ d (m) . nd n =m
r ,d
(15)
d
It is obvious that estimate (15) implies the global boundedness of the sum S(f , p ) for any compactly supported coordinate-wise slow sequence f , f S d ≤ 1: r ,d
sup |S(f , p )| 1.
p
∈P r , d
Estimate (15) also implies the convergence of the infinite series in the Pringsheim sense if f is not a compactly supported sequence. Moreover, the following estimate holds for the convergence rate of the sequence of rectangular partial sums: fn Bn ( p), µ = µ(m) := min mk , Sm (f , p ) := k
n∈m
r ,d (µ) . |S(f , p ) − Sm (f , p )| min 1, max p∗ −ρ k 1≤k≤d
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Case (3). Put A = {1} . In this case we must consider the simple discrete trigonometric sum fn An (p). S(f , p) = S(f , {1}, p) := n∈N
As was noted above, by applying the Abel transform for the partial sum of a series, we obtain Sm (f , p) :=
m
fn An (p) =
n=1
m
fn (Tn (p) − Tn−1 (p)) = fm Tm (p) +
n=1
m−1
Tn (p)∆fn .
n=1
Hence, by Theorem 2 and estimate (a) from Eq. (4), in this case it suffices to prove that if f is a slow sequence, then ∞ |Tn (p)∆fn | < ∞. (17) sup p∈P r n=1
Although the substance of the proof of this assertion is the same as for (a) from Eq. (4) of the paper [4], we will present the most important details here. They also will be required for the complete proof of our theorem, and perhaps they will be interesting for the reader. Let us start with a general characterization of the construction. For a given polynomial p ∈ P r , the domain of summation N is split into two disjoint sets. Each of these sets is the union of pairwise disjoint intervals of integers:
N1 (p) =
[µj , νj ],
N = N1 (p) ∪ N2 (p), N2 (p) = (νj , µj+1 ),
j≥1
µj = µj (p),
νj = νj (p).
(18)
j≥1
This partition is performed in accordance with the arithmetical properties of the vector x of coefficients of the polynomial p/(2π) = P (x, · ) . More precisely, the partition depends on an approximation of x by vectors with rational coordinates. For n ∈ [µj , νj ] , the point x is located close to a rational point yj with “relatively small denominator.” For the trigonometric sums Tn and An here, the following asymptotic formulas hold: pj ) + εn , An (p) = σj Bn ( pj ) + (εn − εn−1 ), n ∈ [µj , νj ], Tn (p) = σj In ( r r |σj | 1, p j ∈ P r , εn n−α , α = α(r) > 0. σj = σj (x),
(19)
j
On the contrary, if n ∈ N2 (p) , then x is far from rational vectors with small denominators. For such n , the following estimates hold: Tn (p) = εn ,
r
|εn | n−β ,
An (p) = εn − εn−1 ,
β = β(r) > 0.
(20)
Suppose that this construction is possible. Then the sum (17) can be estimated by using Eqs. (19) and (20) as follows:
r
|In ( pj )∆fn | ≤
n∈[µj ,νj ]
|Tn (p)∆fn | =
n
sup
p ∈P r n∈N
j
r ,α,β
|Tn (p)∆fn | +
n∈N1 r
r
|In ( p)∆fn | 1 ;
|σj |
|Tn (p)∆fn |
n∈N2
|In ( pj )∆fn | +
n∈[µj ,νj ]
(n
−α
+n
−β
(21) )∆fn
n
r ,α,β
|σj | + 1 1.
j
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For a specific implementation of this construction, we use the Hardy–Littlewood–Vinogradov disk method. This method was developed in order to derive asymptotic formulas and estimates of the Weyl trigonometric sums En (p) = En (x) :=
n n 1 ip(t) 1 2πiP (x, t) e = e . n t=1 n t=1
If one of the coordinates x1 , . . . , xr of the vector x is an irrational number, then by the Weyl Theorem [14], we have lim En (p) = 0. n→∞
For a rational point y ∈ Zr /N , we denote by Q = Q(y) the least common multiple of the denominators of the coordinate in the noncancelable representation and rewrite y in the form y = a/Q:
br b1 , ... , q1 qr
b ∈ Zr , q ∈ Nr , ∈ Zr /Nr , ar a a1 , ... , = , Q = Q(y) := [q1 , . . . , qr ], y = Q Q Q y=
(bs , qs ) = 1, a ∈ Zr ,
Q ∈ N.
For such a rational y , we denote by σ(y) the corresponding (normed) complete rational trigonometric sum (the Gauss sum of maximal order), i.e., σ(y) :=
Q Q 1 2πi(a1 n+···+ar nr )/Q 1 2πiP (y,n) e = e , Q n=1 Q n=1
Q = Q(y).
We have lim En (p) = EQ (p) = σ(y),
n→∞
p = 2πP (y, · ),
and for σ(y) , the following estimates due to Loo-Keng Hua are valid (for the proof see [15, 16]): r
|σ(y)| Q−ρ (y),
ρ=
1 . r
(22)
As was done in [4], we apply Arkhipov’s results [17] in the disk method. In this way, for a given integer n , the space Rr is split into two sets Rr = En ∪ Fn . The coefficients vector x ∈ Rr of the polynomial p = P (x, · )/(2π) ∈ P r belongs to En (or the large arc), if in a small rectangular neighborhood of x a rational point y = (y1 , . . . , yr ) with small denominator Q(y) exists: as s s Q = Q(y) ≤ n0.3 . (23) max n |xs − ys | = max n xs − ≤ n0.3 , 1≤s≤r 1≤s≤r Q All the points x ∈ Rr which do not possess this property form, by definition, the set Fn (or the small arc). The necessary elements of the construction (18)–(20) are introduced by the following assertions. Lemma 2 (Arkhipov [17, Lemmas 7 and 6]). (1) If x ∈ En and y is a rational point satisfying (23), z := x − y , then r 1 n ip(t) e dt + ε, p := 2πP (z, · ), |ε| ≤ 9rQn−1 n−0.7 . (24) En (p) = σ(y) n 0 MATHEMATICAL NOTES
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K. I. OSKOLKOV
(2) If x ∈ Fn , then r
|En (p)| n−β ,
β≥
8r 2 . ln r + 1.5 ln ln r + 4.2
(25)
Next, it is obvious that p(−t) = 2π
r (−1)s xs ts = 2πP (x , t),
x := (−x1 , . . . , (−1)r xr ).
s=1
Moreover, let us note that the following relations, which are important for estimating discrete sums, are valid: σ(y) = σ(y ),
y = (y1 , . . . , yr ) ∈ Zr /Nr ,
y := (−y1 , . . . , (−1)r yr ).
Hence the following assertion is a corollary of Lemma 2. Lemma 3. (1) If x ∈ En , y is a rational point satisfying Eq. (23), and p := 2πP (x − y, · ) , then r
|εn | n−α ,
p) + εn , Tn (p) = σ(y)In (
α = 0.7.
(26)
(2) If x ∈ Fn , then (see also (25)) r
|Tn (p)| n−β .
(27)
Lemma 4 [4]. If n > n0 := 1024 , then for any x ∈ En , the rational point y = a/Q satisfying condition (23) is unique. Proof. If we assume the converse, then another rational point y = a /Q = a/Q satisfying as s max Q , max n xs − ≤ n0.3 1≤s≤r Q exists. Since y = y , it follows that there is an s ∈ [1, r] such that as /Q = as /Q , because as as 1 ≤ − ≤ 2n0.3−s ≤ 2n−0.7 , max(Q, Q ) ≤ n0.3 . QQ Q Q If n > 1024 , then the last two estimates are in contradiction. This proves the uniqueness of y for n > n0 . Let us fix the polynomial p = 2πP (x, · ) ∈ P r . As n ≥ n0 increases, the point x visits En and Fn in turn. According to these events, we split the set N into two subsets N1 (p) := {n ≥ n0 , x ∈ En },
N2 (p) := [1, n0 ) ∪ {n ≥ n0 , x ∈ Fn }.
(28)
Consider the collection of different rational points Y(x) := {y1 , y2 , . . . } in Rr which are sequential rational approximations y = y(n) (x) of the vector x according to Eq. (23) as n increases. Thus, N1 (p) is the union of disjoint intervals of integers [µj , νj ](p):
aj [µj , νj ](p), [µj , νj ](p) := n ≥ n0 , y(n) (x) = yj = . (29) N1 (p) = Qj j≥1
According to this definition, we set N2 (p) = N \ N1 (p) . MATHEMATICAL NOTES
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Lemma 5 (see also [4], Lemma 3). Each pair of subsequent intervals [µj , νj ] , [µj+1 , νj+1 ] in Eq. (29) is “widely separated,” and the denominators Qj “grow rapidly”: 4/3
µj+1 ≥ (0.5)10/3 νj
4/3
Qj+1 ≥ 0.5Qj .
,
(30)
Proof. The proof is analogous to that of Lemma 4. Since yj = yj+1 , it follows that an s ∈ [1, r] exists such that as,j a 1 s,j+1 ≤ ν −0.7 + ν −0.7 ≤ 2ν −0.7 . ≤ − j j+1 j Qj Qj+1 Qj Qj+1 0.3 Hence (µj µj+1 )0.3 ≥ Qj Qj+1 ≥ 0.5νj0.7 , because Qj ≤ µ0.3 j , Qj+1 ≤ µj+1 . This completes the proof of estimates (30).
We conclude that the asymptotic formulas (19) and the estimates (20) hold with σj = σ(yj ) , and Eqs. (30) and (22) imply that r r |σj | Q−ρ j 1. j
j
This proves case (3). As to the convergence rate of the sequence of partial sums Sm (f , p) , this situation is obviously more complicated than the previously considered cases (see (16) and (10)). The convergence rate depends on the arithmetical properties of the coefficients vector x . For a positive number ε in the ε-neighborhood of the point x , we can find a rational vector y = x with the least denominator Q(y) and denote this denominator by Q(p, ε) . It is obvious that Q(p, ε) → ∞ , ε → 0 . Then the previously described construction implies that r
|S(f , p) − Sm (f , p)| Q−ρ (p, m−0.3 ) + m−β . For more details see [9]. Case (4). Again, under the preliminary assumption that the sequence f = {fn } is compactly supported, we shall prove the estimate (5). Let us use induction on the number of elements a := #A of the set A and apply Telyakovskii’s key idea, i.e., split the domain of summation into polynomial octants. If a = 0 , i.e., A = ∅ , then Proposition (5) holds due to the previously considered case (2). Note that if we consider An (p) = Tn (p) − Tn−1 (p) as a function of the coefficients of the polynomial p for fixed n , then this function is periodic in each of these variables with period 2π . Hence we can assume without loss of generality that the coefficients of the polynomial pk in Ank (pk ) do not exceed π in absolute value. Using the induction principle, we suppose that r ,d
sup |S(f , A, p )| 1,
#A = a,
(31)
p
∈P r , d
and derive from this assumption an upper bound for sums over the octants ωm , m ∈ N (see (11)): fn Cn (A, p ), #A = a + 1. Sωm (f , A, p ) := n∈ωm
Rewrite the sums as in [14]:
) = Sωm (f , A, p
d−1 n1 ∈ωm
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Cn1 ( p )Dn1 (f , A, p),
(32)
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K. I. OSKOLKOV
where d
Cn1 (A, p ) :=
Dn1 (f , A, p) :=
Cnk (A, pk ),
fn Cn (A, p).
n : p∗ (n)≥p∗ 2 (n2 )
k=2
By using Eq. (6), we can write a trivial estimate for the product Cn1 (A, p ) (see also Eq. (13)): d
|Cn1 (A, p )|
d min(1, p∗ k (nk )) = Πn1 ( p ). nk
(33)
k=2
As about the sums Dn1 (f , p) , we see that only two possibilities exist. (i) 1 ∈ / A , hence Cn (A, p) = Bn (p) . In this case, we can get the estimates for Dn1 (f , p) and Sωm (f , A, p ) by applying Eqs. (33) and (15) without using the induction hypothesis (31): |Dn1 (f , A, p)| =
n : p∗ (n)≥p∗ 2 (n2 )
r ,d
|Sωm (f , A, p )|
r fn Bn (p) min 1, p∗ −ρ 2 (n2 ) ,
r ,d −ρ Πn1 ( p ) min 1, p∗ −ρ 2 (n2 ) min 1, p∗ d (m) .
(34)
d−1 n1 ∈ωm
(ii) 1 ∈ A , hence Cn (A, p) = An (p) and
Dn1 (f , A, p) =
fn An (p) =
n : p∗ (n)≥p∗ 2 (n2 )
fn Tn (p) − Tn−1 (p) .
(35)
n : p∗ (n)≥p∗ 2 (n2 )
Here we must apply the assumption that the coefficients of the polynomial p do not exceed π in absolute value. Therefore, the inequality p∗ (n) ≥ p∗ 2 (n2 ) implies that n p∗ ρ2 (n2 ),
n−β min(1, p∗ −γ 2 (n2 )),
γ := ρβ.
(36)
Consider a number sequence {εn } satisfying the condition |εn | n−β and the sequence f = {fn } ∈ S , f S ≤ 1 . By applying the Abel transform (see (8)), we see that β fn (εn − εn−1 ) = εm−1 fm − εn ∆fn m−β , n≥m
n≥m
and hence from Eq. (36) it follows that −γ f (ε − ε ) n n n−1 min(1, p∗ 2 (n2 )) n : p∗ (n)≥p∗ 2 (n2 )
By using exactly the same arguments as in the previous case, we obtain from the above estimate the following inequality: Cn1 (A, p ) fn (εn − εn−1 ) d−1 n1 ∈ωm
r ,d
n : p∗ (n)≥p∗ 2 (n2 )
r ,d −γ Πn1 ( p ) min 1, p∗ −γ 2 (n2 ) min 1, p∗ d (m) .
d−1 n1 ∈ωm
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ON A RESULT OF TELYAKOVSKII AND MULTIPLE HILBERT TRANSFORMS
243
By using this estimate, we “sweep out” the summation of residual terms εn in the asymptotical formulas (26) and also the summation in n ∈ N2 (p) . Up to a uniformly bounded error, the summation in n localizes on the major terms in the asymptotic formulas on the set N1 (p) = j [µj , νj ](p) . From Eqs. (26) and (27), we see that r ,d (j ,m) Sω (f , A, p ) − σj S(f ,p j ) min 1, p∗ −γ d (m) , m
(37)
j
where pj , p2 , . . . , pd ), pj = ( fn(j ,m) :=
S(f (j ,m) , pj ) =
fn(j ,m) Bn ( pj )Cn1 ( p ),
n∈Nd
fn
for n ∈ ωm , n ∈ [µj , νj ](p),
0
for of all remaining n ∈ Nd .
For any fixed pair m , j ∈ N , the set {n : n ∈ ωm , n ∈ [µj , νj ](p)} is coordinate-wise convex (j ,m) }n∈Nd is coordinate-wise slow and, (or possibly empty); hence the sequence f (j ,m) := {fn (j ,m)
d ≤ 2 f S d ≤ 2 . Now we see that the total number of factors of type A in moreover, f S the product B dk=2 C is one less than in the original product dk=1 C . Hence, by the induction hypothesis (31), we see that r ,d
|S(f (j ,m) , pj )| 1.
(38)
r ,d r )| 1 , #A = a + 1 . Since we also have j |σj | 1 , Eq. (37) implies that |Sωm (f , A, p This justifies the global boundedness of the total sum for compactly supported sequences f :
sup |S(f , A, p )| < ∞.
p
∈P r , d
In conclusion, let us prove the convergence of the series S(f , A, p ) for slow sequences f which are not compactly supported. To this end, it suffices to show that sup |Sωm (f , A, p )| → 0,
f ∈S0d
m → ∞,
for any fixed set of polynomials p ∈ P r ,d , where S0d denotes the set of all slow compactly supported sequences. We can assume that none of the polynomials pk is identically zero, because the righthand sides of Eq. (34) (case (i)) and Eq.(37) (case (ii)) are uniformly small over S0d . In particular, we can restrict ourself to case (ii). Suppose that one of the coordinates of the coefficients vector x of the polynomial P (x, · ) = p( · )/(2π) is an irrational number. Then for any fixed j , the set {n ∈ ωm , n ∈ [µj , νj ](p)} is empty for all sufficiently large m , so that f (j ,m) ≡ 0,
m ≥ M (j).
Indeed, the condition n = (n, n2 , . . . , nd ) ∈ ωm implies by definition (11) of the octant ωm that p∗ (n) ≥ p∗ d (m) . This contradicts the estimate n ≤ νj (p) if m is sufficiently large. Thus, there exists a sequence {J(m)}m∈N such that J(m) → ∞, MATHEMATICAL NOTES
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j ≤ J(m),
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K. I. OSKOLKOV
and therefore, by Eq. (37), r ,d
Sωm (f , A, p )
p∗ −γ d (m)
+
σj
→ 0,
m → ∞.
j>J(m)
Finally, if x ∈ Zr /N , then the last component of the union N1 (p) = j [µj , νj ] is the semiaxis pJ ) ≡ 0 . Therefore, if m is sufficiently [µJ , ∞) and p J ≡ 0 ; hence for n ≥ µJ we simply have Bn ( j ) = 0 for all j , and for all such m we have large, then S(f (j ,m) , p r ,d
Sωm (f , A, p ) p∗ −γ d (m). This completes the proof of Lemma 1. ACKNOWLEDGMENTS This research was supported by the NSF under grant DMS-9706883. REFERENCES 1. S. A. Telyakovskii, “On estimates of derivatives of trigonometric polynomials of several variables,” Sibirsk. Mat. Zh. [Siberian Math. J.], 4 (1963), no. 6, 1404–1411. 2. S. A. Telyakovskii, “The uniform boundedness of trigonometric polynomials of several variables,” Mat. Zametki [Math. Notes], 42 (1987), no. 1, 33–39. 3. S. A. Telyakovskii and V. N. Temlyakov, “On the convergence of Fourier series of bounded variation in several variables,” Mat. Zametki [Math. Notes], 61 (1997), no. 4, 583–591. 4. G. I. Arkhipov and K. I. Oskolkov, “On a special trigonometric series and its applications,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 134(176) (1987), no. 2(10), 147–157. 5. I. M. Vinogradov, Method of Trigonometric Sums in Number Theory [in Russian], Nauka, Moscow, 1971. 6. E. Stein and S. Wainger, “Discrete analogues of singular Radon transforms,” Bull. Amer. Math. Soc., 23 (1990), 537–544. 7. K. I. Oskolkov, “The class of Vinogradov’s series and its applications in harmonic analysis,” in: Progress in Approximation Theory, An International Prospective Proceedings of the International Conference on Approximation Theory held on March 19–22, 1990, Springer-Verlag, 1992, pp. 353–402. 8. K. I. Oskolkov, “Schr¨ odinger equation and oscillatory Hilbert transforms of second degree,” J. Fourier Anal. Appl., 4 (1998), 341–356. 9. K. I. Oskolkov, “Vinogradov series in the Cauchy problem for the Schr¨ odinger equation,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 200 (1991), 265–288. 10. A. Zygmund, Trigonometric Series, Second Edition, Cambridge University Press, Cambridge, 1959. 11. M. Z. Garaev, “On a multiple trigonometric series,” Acta Arithm., 102 (2002), no. 2, 183–187. 12. E. Stein and S. Wainger, “The estimation of an integral arising in multiplier transformations,” Studia Math., 35 (1970), 101–104. 13. E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. ¨ 14. H. Weyl, “Uber die Gleichverteilung von Zahlen mod Eins,” Math. Ann., 77 (1916), 313–352. 15. Chen. Jing-run, “On Professor Hua’s estimate of exponential sum.,” Sci. Sinica, 20 (1977), 711–719. 16. S. B. Stechkin, “An estimate of a complete rational trigonometric sum,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 143 (1977), 188–207. 17. G. I. Arkhipov, “On the Hilbert–Kamke problem,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 48 (1984), no. 1, 3–52. University of South Carolina, Columbia, USA
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