Asymptotics of the Lebesgue constants for bivariate approximation processes
aa r X i v : . [ m a t h . C A ] S e p Asymptotics of the Lebesgue constants for bivariateapproximation processes
Yurii Kolomoitsev a, b, *, 1 and Tetiana Lomako b Abstract.
In this paper asymptotic formulas are given for the Lebesgue constants gen-erated by three special approximation processes related to the ℓ -partial sums of Fourierseries. In particular, we consider the Lagrange interpolation polynomials based on theLissajous-Chebyshev node points, the partial sums of the Fourier series generated by theanisotropically dilated rhombus, and the corresponding discrete partial sums.
1. Introduction
Let D ⊂ R d be a compact set with non-empty interior. Denote by C ( D ) the set of allreal-valued continuous functions on D equipped with the norm k f k = sup x ∈ D | f ( x ) | . By Π n , n = ( n , . . . , n d ) ∈ N d , we denote some abstract space of polynomial functions in C ( D ) (e.g.,algebraic or trigonometric polynomials of degree n in some sense).Consider a projection operator P n : C ( D ) Π n . The norm of this operator Λ n = k P n k = sup k f k≤ k P n ( f ) k is called the Lebesgue constant of P n . In view of the well-knowninequality k f − P n ( f ) k ≤ (1 + Λ n ) inf P ∈ Π n k f − P k , the Lebesgue constant is an essential tool for the investigation of approximation propertiesof the operator P n . We are interested in studying the Lebesgue constants in the case P n isan interpolation process or a partial sum of the Fourier series.For univariate interpolation processes, the behaviour of the Lebesgue constants is wellstudied for the most classical sets of nodes. For example, if P n is the Lagrange interpolationpolynomial with the Chebyshev nodes on [ − , n = 2 π log n + 2 π (cid:18) γ + log 8 π (cid:19) + O (cid:18) n (cid:19) , where γ denotes Euler’s constant, see, e.g., [ , p. 65]. See also [ ] for related asymptoticsof the Lebesgue constant of interpolation processes based on other sets of nodes.In multivariate spaces, much less is known except the trivial case of the Lagrange in-terpolation based on the tensor product grid. One of the main problems is the choice of asuitable set of nodes. At the present time, promising Lagrange interpolation polynomials on Mathematics Subject Classification.
Key words and phrases.
Lebesgue constants, asymptotic formula, anisotropy, Dirichlet kernel, interpola-tion, Lissajous-Chebyshev nodes. a Universit¨at zu L¨ubeck, Institut f¨ur Mathematik, Ratzeburger Allee 160, 23562 L¨ubeck, Germany. b Institute of Applied Mathematics and Mechanics of NAS of Ukraine, General Batyuk Str. 19, Slov’yans’k,Donetsk region, Ukraine, 84100. Supported by DFG project KO 5804/1-1. ∗ Corresponding author: [email protected]. [ − , are constructed by means of the so-called Padua points Pad n , see, e.g., [ ]. Notethat these polynomials have a series of favourite properties, one of which is that the Paduapoints can be characterized as a set of node points of a particular Lissajous curve (see [ ]).The Lebesgue constant of the Lagrange interpolation at the Padua points was studied in [ ].In particular, it was shown that it grows like O (log n ), where n is total degree of the cor-responding polynomial space Π n . Similar result for the Lebesgue constat of the Lagrangeinterpolation based on the Xu points (see [ ]) was early obtained in [ ]. The estimates frombelow for different Lebesgue constants were investigated in [
6, 24, 25 ]. The correspondingresults imply that for the mentioned interpolation processes Λ n ≥ c log n .In this paper, we study the Lebesgue constant for the interpolation processes based onthe so-called Lissajous-Chebyshev nodes LC n , which represent an anisotropic analogue of thePadua points in the sense that LC n,n +1 = Pad n , see [ ]. Different properties of the polynomialinterpolation on the Lissajous-Chebyshev nodes have been recently obtained in [
5, 6, 7, 8,14 ]. Note that the study of interpolation on such nodes is motivated by applications in a novelmedical imaging technology called Magnetic Particle Imaging (see, e.g., [ ]). Sharp estimatesof the Lebesgue constant of the interpolation processes on LC n were recently established in [ ].In particular, it was shown that in the multivariate case c ( d ) d Y j =1 log n j ≤ Λ LC n ≤ c ( d ) d Y j =1 log n j . The main goal of this paper is to find an asymptotic formula for Λ LC n in the two dimen-sional case. Our approach is similar to one given in [ ]. First, we investigate the Lebesgueconstants for the ℓ -partial sums of Fourier series and then establish relationships betweenthe corresponding Lebesgue constants. For our purposes, we essentially extend a series ofequalities and estimates obtained in [ ] and [ ] for the Dirichlet kernels with the frequen-cies in the rhombus { ( k , k ) : | k | /n + | k | /n ≤ } and apply new methods developedrecently in the papers [ ] and [ ].It is worth noting that the study of the Lebesgue constants for the partial sums of Fourierseries has its own great interest. There are many works dedicated to this topic (see, e.g.,surveys [ ] and [ ]). Our investigation concerns the Lebesgue constants for polyhedral par-tial sums of Fourier series. In particular, we are interested in the so-called triangular partialsums, which is also called the ℓ -partial sums. The estimates for the Lebesgue constantsof such partial sums were obtained, e.g., in [
1, 6, 12, 13, 16, 20, 25 ]. Different asymp-totic formulas for the Lebesge constant in the case of isotropic dilations of a polyhedra weregiven in [
15, 20, 21, 22 ]; for the anisotropic case see [ ] and [ ] as well as the recentwork [ ], in which an asymptotic formula is given for the Lebesgue constants generated bythe anisotropically dilated d -dimensional simplex.Throughout the paper, we use the following notation: T d = [ − π, π ) d , T = T , and k f k L (Ω) = Z Ω | f ( x ) | d x . For any m, n ∈ N , we denote x µ = x ( m ) µ = πµ/m , y ν = y ( n ) ν = πν/n , and k ( a µ,ν ) µ,ν k ℓ mn = 1 mn m − X µ =0 n − X ν =0 | a µ,ν | . The floor and the fractional part functions are defined, as usual, by [ x ] = max( m ∈ Z : m ≤ x ) and { x } = x − [ x ] , correspondingly. Also, we use the notation F . G, with F, G ≥
0, forthe estimate F ≤ c G, where c is an absolute positive constant. SYMPTOTICS OF THE LEBESGUE CONSTANT 3
2. Main results2.1. Polynomial interpolation on Lissajous-Chebyschev nodes.
Let N = { ( m, n ) ∈ N : m and n are relatively prime } . In what follows, for simplicity, we use thenotation u k = cos (cid:18) kπm (cid:19) , v l = cos (cid:18) lπn (cid:19) , k = 0 , . . . , m, l = 0 , . . . , n, to abbreviate the Chebyshev-Gauss-Lobatto points. For each ( m, n ) ∈ N , the Lissajous-Chebyshev node points of the (degenerate) Lissajous curve γ mn ( t ) = (cos( nt ) , cos( mt )) aredefined by LC mn = (cid:26) γ mn (cid:18) πkmn (cid:19) , k = 0 , . . . , mn (cid:27) . Figure 1
Illustration of the degenerate Lissajous curve γ , and the set LC , .Note that (see [ ]) for every ( m, n ) ∈ N , the set LC mn contains ( m + 1)( n + 1) / mn = { ( u i , v j ) : ( i, j ) ∈ I mn } , where I mn = (cid:8) ( i, j ) ∈ Z : i = 0 , . . . , m, j = 0 , . . . , n, i + j = 0 (mod 2) (cid:9) . Let C n be the normalized Chebyshev polynomial defined by C n ( u ) = (cid:26) , n = 0, √ n arccos u ) , n = 0.A proper set of polynomials for interpolation on LC mn is given byΠ mn = span { C k ( u ) C l ( v ) : ( k, l ) ∈ Γ mn } , where Γ mn = (cid:26) ( i, j ) ∈ Z : im + jn < (cid:27) ∪ { (0 , n ) } . It was proved in [ ] that for every continuous function f : [ − , → R , the uniquesolution of the interpolation problem P mn ( f )( u k , v l ) = f ( u k , v l ) , ( u k , v l ) ∈ LC mn , YURII KOLOMOITSEV AND TETIANA LOMAKO in the space Π nm is given by the polynomial P mn ( f )( u, v ) = X ( k,l ) ∈I mn f ( u k , v l ) ϕ mn ( u, v ; u k , v l ) , where ϕ mn ( u, v ; u k , v l ) = l kl (cid:18) X ( i,j ) ∈ Γ mn C i ( u k ) C j ( v l ) C i ( u ) C j ( v ) − C n ( v l ) C n ( v ) (cid:19) and l kl := / (2 mn ) , ( u k , v l ) is a vertex point of [ − , ,1 / ( mn ) , ( u k , v l ) is an edge point of [ − , ,2 / ( mn ) , ( u k , v l ) is an interior point of [ − , .Consider the Lebesgue constant of the above interpolation problem. Note that it can alsobe defined by Λ LC mn := max ( u,v ) ∈ [ − , X ( k,l ) ∈I mn | ϕ mn ( u, v ; u k , v l ) | . The following theorem is our main result.
Theorem . Let m, n ∈ N be such that n = lm + p , where l, p ∈ N , ≤ p < m . Then Λ LC mn = 4 π (2 log m log n − log m ) + O (cid:18) log n + p log mp (cid:19) . In particular, Theorem 2.1 implies that for the interpolation based on the Padua pointsPad n , we have Λ Pad n = 4 π log n + O (log n ) . Similar formula holds for Λ LC mn if n ∼ m (i.e., l and p are certain fixed numbers). At the sametime, if log n log m → ∞ as m → ∞ , thenΛ LC mn = 8 π log n log m + O (log n + log m ) . This formula follows from the proof of Theorem 2.1 given in the last section and inequal-ity (3.12). ℓ -partial sums of Fourier series. Let f be an integrable 2 π -periodic in eachvariable function. The Fourier series of f is given by f ( x, y ) ∼ X k,l ∈ Z c kl ( f ) e i ( kx + ly ) , where c kl ( f ) = 14 π Z T f ( x, y ) e − i ( kx + ly ) dxdy. In the multivariate case, there are many ways to define the partial sums of Fourier series (see,e.g., [ ]). In this paper, we consider the so called ℓ -partial sums:(2.1) S mn ( f )( x, y ) = X | k | /m + | l | /n ≤ c kl ( f ) e i ( kx + ly ) . The Lebesgue constant of this partial sums is denoted by L mn := kS mn k C ( T ) → C ( T ) = sup k f k≤ |S mn ( f )( x, y ) | . SYMPTOTICS OF THE LEBESGUE CONSTANT 5
It is well known that(2.2) L mn = 14 π Z T | D mn ( x, y ) | dxdy, where D mn ( x, y ) = X | k | /m + | l | /n ≤ e i ( kx + ly ) is the corresponding Dirichlet kernel.The following asymptotic equality was established in [ ]:(2.3) L mn = 16 π (2 log m log n − log m ) + O (log n ) , where m, n ∈ N are such that nm ∈ N .We improve this formula by considering arbitrary m, n ∈ N . Theorem . Let m, n ∈ N , ≤ m ≤ n . Then (2.4) L mn = 16 π (2 log m log n − log m ) + k F mn k L ( T ) + O (log log n log m + log n ) , where F mn ( x, y ) := 4 m X k =1 n − nm k o cos kx cos n (cid:18) km (cid:19) y. (2.5) Moreover, if m and n are such that n = lm + p , where l, p ∈ N , ≤ p ≤ m . Then (2.6) L mn = 16 π (2 log m log n − log m ) + O (cid:18) log n + p log mp (cid:19) . Remark . It is known that k F mn k L ( T ) = O (log m ) , see Lemma 3.2 below. In somespecial cases, one can even show that this estimate is sharp, see, e.g., [ ] . At the same time,it follows from Lemma 3.3 that in the case n = lm + p , where l, p ∈ N , one has the estimate k F mn k L ( T ) = O ( p log mp ) , which is better than log m for an appropriate fixed p . Generally,asymptotic properties of k F mn k L ( T ) are unknown, see, e.g., [ ] for a discussion. ℓ -discrete partial sums of Fourier series. The discrete analogue of partialsums (2.1) is given by e S mn ( f )( x, y ) = X | k | /m + | l | /n ≤ e c kl ( f ) e i ( kx + ly ) , where e c kl ( f ) = 14 mn X − m ≤ µ ≤ m − − n ≤ ν ≤ n − f ( x µ , y ν ) e − i ( kx µ + ly ν ) . The Lebesgue constant of e S mn can be defined byΛ mn := sup ( x,y ) ∈ [0 ,π ] L mn ( x, y ) , where L mn ( x, y ) := sup k f k≤ | e S mn ( f )( x, y ) | = 14 mn X − m ≤ µ ≤ m − − n ≤ ν ≤ n − | D mn ( x − x µ , y − y ν ) | is the corresponding Lebesgue function.The following asymptotic equality was obtained in [ ]: L mn ( x, y ) = 2 π Φ mn ( x, y ) (cid:0) m log n − log m (cid:1) + O (log n ) , YURII KOLOMOITSEV AND TETIANA LOMAKO where m, n ∈ N are such that nm ∈ N andΦ mn ( x, y ) := (cid:12)(cid:12)(cid:12) sin mx + ny mx − ny (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) cos mx + ny mx − ny (cid:12)(cid:12)(cid:12) . (2.7)In particular,(2.8) Λ mn = 2 π (cid:0) m log n − log m (cid:1) + O (log n ) . In the next theorem, we obtain an analogue of (2.8) for any m and n . Theorem . Let m, n ∈ N , ≤ m ≤ n . Then Λ mn = 2 π (cid:0) m log n − log m (cid:1) + F mn + O (log log n log m + log n ) , (2.9) where F mn := sup ( x,y ) ∈ T mn X ≤ µ ≤ m − ≤ ν ≤ n − | F mn ( x + x µ , y + y ν ) | and F mn is defined in (2.5) . Moreover, if m and n are such that n = lm + p , where l, p ∈ N , ≤ p ≤ m . Then Λ mn = 2 π (cid:0) m log n − log m (cid:1) + O (cid:18) log n + p log mp (cid:19) . (2.10) Remark . The same comments as in Remark 2.1 hold true for the quantity F mn . Tosee this, it suffices to apply the Marcinkiewicz-Zygmund inequality (see Lemma 3.4), whichshows that F mn . sup y ∈ T k F mn ( · , y ) k L ( T ) , and then to use Lemma 3.3.
3. Auxiliary results
For any m, n ∈ N , we denote S mn ( x, y ) := 2 y D m (cid:16) x + nm y (cid:17) D m (cid:16) x − nm y (cid:17) sin nm y, (3.1)where D m ( x ) := sin mx sin x , and R mn ( x, y ) := X ν =0 yπν (2 πν + y ) m X k = − m e ikx sin (cid:18) n (cid:18) − | k | m (cid:19) (2 πν + y ) (cid:19) + m X k = − m e ikx cos (cid:18) n (cid:18) − | k | m (cid:19) y (cid:19) . (3.2) Lemma . For any m, n ∈ N , we have (3.3) D mn ( x, y ) = S mn ( x, y ) − F mn ( x, y ) + R mn ( x, y ) , where F mn is defined in (2.5) . SYMPTOTICS OF THE LEBESGUE CONSTANT 7
Proof.
We obtain D mn ( x, y ) = m X k = − m e ikx [ n (1 − | k | m )] X l = − [ n (1 − | k | m )] e ily = m X k = − m e ikx Z n (1 − | k | m ) − n (1 − | k | m ) e ily d [ l ]= m X k = − m e ikx Z n (1 − | k | m ) − n (1 − | k | m ) e ily dl − m X k = − m e ikx Z n (1 − | k | m ) − n (1 − | k | m ) e ily d { l } := J ( x, y ) − J ( x, y ) . Consider J . We have J ( x, y ) = 2 y m X k = − m e ikx sin n (cid:18) − | k | m (cid:19) y = 2 sin nyy + 4 y m X k =1 cos kx sin n (cid:18) − km (cid:19) y. (3.4)Next, using [ , AD(361.8)] and the standard trigonometric identities, we derive2 m X k =1 cos kx sin n (cid:18) − km (cid:19) y = m X k =1 (cid:16) sin (cid:16)(cid:0) x − nm y (cid:1) k + ny (cid:17) − sin (cid:16)(cid:0) x + nm y (cid:1) k − ny (cid:17)(cid:17) = sin (cid:18) m + 12 (cid:0) x − nm y (cid:1) + ny (cid:19) D m (cid:16) x − nm y (cid:17) − sin (cid:18) m + 12 (cid:0) x + nm y (cid:1) − ny (cid:19) D m (cid:16) x + nm y (cid:17) = sin (cid:18) mx + ny (cid:0) x − nm y (cid:1)(cid:19) D m (cid:16) x − nm y (cid:17) − sin (cid:18) mx − ny (cid:0) x + nm y (cid:1)(cid:19) D m (cid:16) x + nm y (cid:17) = sin (cid:18) mx + ny (cid:19) sin (cid:18) mx − ny (cid:19) (cid:18) cot 12 (cid:0) x − nm y (cid:1) − cot 12 (cid:0) x + nm y (cid:1)(cid:19) + sin (cid:18) mx − ny (cid:19) cos (cid:18) mx + ny (cid:19) − cos (cid:18) mx − ny (cid:19) sin (cid:18) mx + ny (cid:19) = D m (cid:16) x + nm y (cid:17) D m (cid:16) x − nm y (cid:17) sin nm y − sin ny. (3.5)Combining (3.4) and (3.5), we get J ( x, y ) = 2 y D m (cid:16) x + nm y (cid:17) D m (cid:16) x − nm y (cid:17) sin nm y = S mn ( x, y ) . (3.6)Consider J . Integrating by parts, we obtain Z n (1 − | k | m ) − n (1 − | k | m ) e ily d { l } = 2 n − nm | k | o cos n (cid:18) − | k | m (cid:19) y − e − in (1 − | k | m ) y − iy Z n (1 − | k | m ) − n (1 − | k | m ) { l } e ily dl. (3.7) YURII KOLOMOITSEV AND TETIANA LOMAKO
Next, representing {·} via the Fourier series as { ξ } = 12 + X ν =0 πiν e πiνξ , we derive Z n (1 − | k | m ) − n (1 − | k | m ) { l } e ily dl = 1 y sin n (cid:18) − | k | m (cid:19) y + X ν =0 πiν Z n (1 − | k | m ) − n (1 − | k | m ) e πiνl e ily dl = 1 y sin n (cid:18) − | k | m (cid:19) y + X ν =0 sin (cid:16) (2 πν + y ) n (cid:16) − | k | m (cid:17)(cid:17) πiν (2 πν + y ) . (3.8)Combining (3.7) and (3.8), we get Z n (1 − | k | m ) − n (1 − | k | m ) e ily d { l } = (cid:16) n − nm | k | o − (cid:17) cos n (cid:18) − | k | m (cid:19) y − y X ν =0 sin (cid:16) (2 πν + y ) n (cid:16) − | k | m (cid:17)(cid:17) πν (2 πν + y ) . This implies that J ( x, y ) = m X k = − m e ikx (cid:16) n − nm | k | o − (cid:17) cos n (cid:18) − | k | m (cid:19) y − X ν =0 yπν (2 πν + y ) m X k = − m e ikx sin (cid:18) (2 πν + y ) n (cid:18) − | k | m (cid:19)(cid:19) = F mn ( x, y ) − X ν =0 yπν (2 πν + y ) m X k = − m e ikx sin (cid:18) (2 πν + y ) n (cid:18) − | k | m (cid:19)(cid:19) − m X k = − m e ikx cos n (cid:18) − | k | m (cid:19) y. (3.9)Finally, combining (3.6) and (3.9), we prove the lemma. (cid:3) Remark . It follows from (3.5) and (3.6) that S mn ( x, y ) = 2 y sin (cid:18) m + 12 (cid:0) x − nm y (cid:1) + ny (cid:19) D m (cid:16) x − nm y (cid:17) − sin (cid:18) m + 12 (cid:0) x + nm y (cid:1) − ny (cid:19) D m (cid:16) x + nm y (cid:17) + sin ny ! . (3.10)Denote by D m the following modification of the Dirichlet kernel D m : D m ( x ) := m X k =1 e ikx . Lemma . Let m, n ∈ N be such that ≤ m ≤ n . Then (3.11) kD m k L ( T ) . log m, (3.12) k F mn k L ( T ) . log m, SYMPTOTICS OF THE LEBESGUE CONSTANT 9 (3.13) k R mn k L ( T ) . log m. Proof.
Inequality (3.11) is well known, see, e.g., [ , Ch. II, § ]. Inequality (3.13) follows easily from (3.2) and (3.11). (cid:3) Denote F mp ( x ) := m X k =0 n − pkm o e ikx . Lemma . Let m, p ∈ N . Then (3.14) kF mp k L ( T ) . log m. Moreover, if ≤ p < m . Then (3.15) kF mp k L ( T ) . p log mp . In the above inequalities, the constant in . does not depend on p and m . Proof.
The proof of (3.14) can be found in [ ] (cf. (3.12)). Let us prove (3.15). Wehave F mp ( x ) = p − X l =0 [ ( l +1) mp ] X k =[ lmp ]+1 (cid:26) − pkm (cid:27) e ikx = p − X l =0 e i [ lmp ] x [ ( l +1) mp ] − [ lmp ] X k =1 (cid:26) − pm (cid:16) k − n lmp o(cid:17)(cid:27) e ikx . (3.16)Since 1 ≤ k ≤ mp − { ( l +1) pm } + { lpm } , we get 0 < pm ( k − { lmp } ) ≤ − pm { ( l +1) pm } <
1, whichtogether with (3.16) implies F mp ( x ) = − p − X l =0 e i [ lmp ] x [ ( l +1) mp ] − [ lmp ] X k =1 pm (cid:16) k − n lmp o(cid:17) e ikx = − pm p − X l =0 e i [ lmp ] x (cid:18) D ′ [ ( l +1) mp ] − [ lmp ] ( x ) − n lmp o D [ ( l +1) mp ] − [ lmp ] ( x ) (cid:19) . Thus, using the Bernstein inequality (see, e.g., [ , Ch. 10, § . 3]) and (3.11), we derive kF mp k L ( T ) ≤ pm p − X l =0 (cid:18)(cid:13)(cid:13)(cid:13) D ′ [ ( l +1) mp ] − [ lmp ] (cid:13)(cid:13)(cid:13) L ( T ) + (cid:13)(cid:13)(cid:13) D [ ( l +1) mp ] − [ lmp ] (cid:13)(cid:13)(cid:13) L ( T ) (cid:19) . p log mp , which proves the lemma. (cid:3) Recall the well-known Marcinkiewicz-Zygmund inequality for trigonometric polynomials(see, e.g., [ , 4.3.3]). Lemma . Let T be a trigonometric polynomial of degree at most n . Then n n − X ν =0 | T ( πν/n ) | . k T k L ( T ) . Lemma . Let m, n ∈ N be such that ≤ m ≤ n . Then (3.17) k D mn k L ([0 ,π ) ) = k S mn k L ([0 ,π ) ) + k F mn k L ([0 ,π ) ) + O (log log n log m ) . If, additionally, | x | ≤ π/ (2 m ) and | y | ≤ π/ (2 n ) , then k ( D mn ( x + x µ , y + y ν )) k ℓ mn = k ( S mn ( x + x µ , y + y ν )) k ℓ mn + k ( F mn ( x + x µ , y + y ν )) k ℓ mn + O (log log n log m ) . (3.18) Moreover, if m and n are such that n = lm + p , p < m , l, p ∈ N . Then (3.19) k D mn k L ([0 ,π ) ) = k S mn k L ([0 ,π ) ) + O (cid:18) log m + p log mp (cid:19) and (3.20) k ( D mn ( x + x µ , y + y ν )) k ℓ mn = k ( S mn ( x + x µ , y + y ν )) k ℓ mn + O (cid:18) log m + p log mp (cid:19) . Proof.
Using (3.14), we have Z π dx Z n | F mn ( x, y ) | dy ≤ Z n dy Z T (cid:12)(cid:12)(cid:12)(cid:12) m X k =1 n − nkm o e ikx (cid:12)(cid:12)(cid:12)(cid:12) dx . log m. This along with (3.3) and (3.13) implies Z π dx Z n | D mn ( x, y ) | dy = Z π dx Z n | S mn ( x, y ) | dy + Z π dx Z n | F mn ( x, y ) | dy + O (log m ) . (3.21)At the same time using (3.10) and (3.11), we get Z π dx Z π n | S mn ( x, y ) | dy . log log n Z π | D m ( x ) | dx . log m log log n. As above, this along with (3.3) and (3.13) implies Z π dx Z π n | D mn ( x, y ) | dy = Z π dx Z π n | S mn ( x, y ) | dy + Z π dx Z π n | F mn ( x, y ) | dy + O (log m log log n ) . (3.22)Thus, combining (3.21) and (3.22), we obtain (3.17).The proof of (3.18) is similar. Using Lemma 3.4 and (3.14), we derive after simplecalculations that1 mn m − X µ =0 [ n log n ] − X ν =0 | F mn ( x + x µ , y + y ν ) | . n [ n log n ] − X ν =0 Z π − π (cid:12)(cid:12)(cid:12)(cid:12) m X k =1 n − nkm o e ikt (cid:12)(cid:12)(cid:12)(cid:12) dt . log m. (3.23) SYMPTOTICS OF THE LEBESGUE CONSTANT 11
Next, using (3.10), (3.11), and Lemma 3.4, we obtain1 mn n − X ν =[ n log n ] m − X µ =0 | S mn ( x + x µ , y + y ν ) |≤ mn n − X ν =[ n log n ] | y + y ν | m − X µ =0 (cid:18)(cid:12)(cid:12)(cid:12) D m (cid:16) x − nm y + x µ − ν (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) D m (cid:16) x + nm y + x µ + ν (cid:17)(cid:12)(cid:12)(cid:12) + | sin n ( y + y ν ) | (cid:19) . n − X ν =[ n log n ] ν Z T (cid:18)(cid:12)(cid:12)(cid:12) D m (cid:16) x − nm y + t − x ν (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) D m (cid:16) x + nm y + t + x ν (cid:17)(cid:12)(cid:12)(cid:12) + 1 (cid:19) dt . log log n log m. (3.24)Thus, combining (3.23) and (3.24) with (3.3) and using the same arguments as in the proofof (3.17), we get (3.18).The proofs of (3.19) and (3.20), follow from equality (3.3), inequalities (3.13),(3.15), and the Marcinkiewicz-Zygmund inequality given in Lemma 3.4, which shows thatsup x,y k ( F mn ( x + x µ , y + y ν )) k ℓ mn . p log mp . (cid:3) We need the following additional notations:∆ (1) m ( x, y ) = 12 S mm ( x, y ) , ∆ (2) m ( x, y ) = D m ( x − y ) sin (cid:18) m ( x + y )2 (cid:19) ,A m = { ( µ, ν ) ∈ Z : µ, ν = 0 , . . . , m − , | µ − ν | ≥ } , and B m = { ( µ, ν ) ∈ A m : µ + ν = 0 (mod 2) } . Lemma . Let m ∈ N , m ≥ . Then k ∆ (1) m k L ([0 ,π ) ) = 8 π log m + O (log m ) , (3.25) k ∆ (2) m k L ([0 ,π ) ) = 16 π log m + O (1) . (3.26) If, additionally, | x | ≤ π/ (2 m ) and | y | ≤ π/ (2 m ) . Then m X A m | ∆ (1) m ( x + x µ , y + x ν ) | = 1 π Φ mm ( x, y ) log m + O (log m ) , (3.27) 1 m X A m | ∆ (2) m ( x + x µ , y + x ν ) | = 2 π Φ mm ( x, y ) log m + O (1) , (3.28) 1 m X B m | ∆ (1) m ( x + x µ , y + x ν ) | = 1 π (cid:12)(cid:12)(cid:12) sin m ( x + y )2 sin m ( x − y )2 (cid:12)(cid:12)(cid:12) log m + O (log m ) , (3.29) 1 m X B m | ∆ (2) m ( x + x µ , y + x ν ) | = 2 π (cid:12)(cid:12)(cid:12) sin m ( x + y )2 sin m ( x − y )2 (cid:12)(cid:12)(cid:12) log m + O (1) , (3.30) where Φ mm is defined in (2.7) . Proof.
Equalities (3.25) and (3.26) can be found in [ ]. The proofs of relations (3.27)–(3.30) are given in [ ]. (cid:3) Lemma . Let m, n ∈ N be such that ≤ m ≤ n . Then (3.31) k S mn k L ([0 ,π ) ) = 16 π (2 log m log n − log m ) + O (log n ) . If, additionally, | x | ≤ π/ (2 m ) and | y | ≤ π/ (2 n ) , then k ( S mn ( x + x µ ,y + y ν )) k ℓ mn = 2 π Φ mn ( x, y )(2 log m log n − log m ) + O (log n ) . (3.32) Proof.
We have k S mn k L ([0 ,π ) ) = Z π dx Z nm π | S mm ( x, y ) | dy = Z π Z π | . . . | + Z π Z [ nm ] ππ | . . . | + Z π Z nm π [ nm ] π | . . . | = I + I + I . (3.33)Using (3.25), we get I = 2 k ∆ (1) m k L ([0 ,π ) ) = 16 π log m + O (log m ) . (3.34)Consider I . Denoting l = [ nm ], we derive I = l − X k =1 Z π dx Z ( k +1) π kπ | S mm ( x, y ) | dy = 2 l − X k =1 Z π Z π | D m ( x + y + πk ) D m ( x − y − πk ) | sin yy + πk dxdy = 2 π l − X k =1 k Z π Z π | D m ( x + y + πk ) D m ( x − y − πk ) | sin y dxdy + Q m , (3.35)where Q m = 2 l − X k =1 Z π Z π | D m ( x + y + πk ) D m ( x − y − πk ) | sin y (cid:18) y + πk − kπ (cid:19) dxdy. Observe that Z π Z π | D m ( x + y + πk ) D m ( x − y − πk ) | sin y dxdy = Z π Z π | D m ( x + y ) D m ( x − y ) | sin y dxdy. (3.36)This together with the equalitysin y = sin x + y x − y − sin x − y x + y | Q m | . log m . Thus, using again (3.36), we derivefrom (3.35): I = 2 π log nm Z π Z π | D m ( x + y ) D m ( x − y ) | sin y dxdy + O (log m ) . (3.38)Next, applying (3.37) and the fact that Z π Z π (cid:12)(cid:12)(cid:12)(cid:12) D m ( x + y ) sin m x − y ) (cid:12)(cid:12)(cid:12)(cid:12) cos x + y dxdy = 0 , SYMPTOTICS OF THE LEBESGUE CONSTANT 13 we obtain Z π Z π | D m ( x + y ) D m ( x − y ) | sin y dxdy = Z π Z π (cid:12)(cid:12)(cid:12)(cid:12) D m ( x − y ) sin m x + y ) (cid:12)(cid:12)(cid:12)(cid:12) cos x − y dxdy = k ∆ (2) m k L ([0 ,π ) ) − Z π Z π | ∆ (2) m ( x, y ) | sin x − y dxdy = k ∆ (2) m k L ([0 ,π ) ) − Z π Z π (cid:12)(cid:12)(cid:12)(cid:12) sin m x + y ) sin m x − y ) (cid:12)(cid:12)(cid:12)(cid:12) tan x − y dxdy = k ∆ (2) m k L ([0 ,π ) ) + O (1) . (3.39)Now, combining (3.38), (3.39) and (3.26), we get I = 32 π log n log m + O (log n ) . (3.40)Consider I . Using (3.10) and estimate (3.11), we derive I . Z π dx Z nm π [ nm ] π ( | D m ( x + y ) | + | D m ( x − y ) | + 1) dy . k D m k L ( T ) . log m. (3.41)Thus, combining (3.33), (3.34), (3.40), and (3.41), we obtain (3.31).Now we prove (3.32). As above, we have k ( S mn ( x + x µ , y + y ν )) k ℓ mn = 1 mn m − X µ =0 m − X ν =0 | . . . | + 1 mn m − X µ =0 m [ nm ] − X ν = m | . . . | + 1 mn m − X µ =0 n − X ν = m [ nm ] | . . . | = J + J + J . (3.42)Here, we suppose that P Bν = A = 0 if A > B . Repeating arguments similar to those made inthe proof of (3.31) (see equalities for I and I ) and using (3.27) and (3.28), we obtain J + J = 2 m X A m (cid:12)(cid:12)(cid:12)(cid:12) ∆ (1) m (cid:16) x + x µ , nm (cid:16) y + y ν (cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) log nm (cid:17) πm X A m (cid:12)(cid:12)(cid:12)(cid:12) ∆ (2) m (cid:16) x + x µ , nm (cid:16) y + y ν (cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + O (log m )= 2 π Φ mn ( x, y )(2 log m log n − log m ) + O (log n ) . To estimate J , we apply (3.10), Lemma 3.4, and (3.11): J . m m − X ν =0 n − X ν = m [ nm ] ν (cid:18)(cid:12)(cid:12)(cid:12) D m (cid:16) x − nm y + x µ − x ν (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) D m (cid:16) x + nm y + x µ + x ν (cid:17)(cid:12)(cid:12)(cid:12) + 1 (cid:19) . log (cid:18) nm h nm i − + 1 (cid:19) k D m k L ( T ) . log m. (3.43)Finally, combining (3.42) and (3.43), we get (3.32). (cid:3)
4. Proofs of the main results
Proof of Theorem 2.1.
Using the substitution u = cos x , v = cos y , and denoting D mn ( x, y ) = X | k | /m + | l | /n< e i ( kx + ly ) , we obtainΛ LC mn = max ( x,y ) ∈ [0 ,π ] X ( µ,ν ) ∈I m,n l µ,ν (cid:12)(cid:12)(cid:12)(cid:12) X ( i,j ) ∈ Γ m,n ix µ cos jy ν cos ix cos jy − cos ny ν cos ny (cid:12)(cid:12)(cid:12)(cid:12) = max ( x,y ) ∈ [0 ,π ] X ( µ,ν ) ∈I m,n l µ,ν (cid:12)(cid:12)(cid:12)(cid:12) X ( i,j ) ∈ Γ m,n ix µ cos jy ν cos ix cos jy (cid:12)(cid:12)(cid:12)(cid:12) + O (1)= max ( x,y ) ∈ [0 ,π ] X ( µ,ν ) ∈I m,n l µ,ν |D mn ( x µ + x, y ν + y ) + D mn ( x µ + x, y ν − y )+ D mn ( x µ − x, y ν + y ) + D mn ( x µ − x, y ν − y ) | + O (1) ≤ max ( x,y ) ∈ [0 ,π ] X ( µ,ν ) ∈I m,n l µ,ν |D mn ( x µ + x, y ν + y ) | + O (1) . (4.1)Denote by LC ′ mn a subset of LC mn , which consists of vertex and edge points of [ − , . Itwas shown in [ ] thatLC ′ mn = ( u i , v j ) : i = 1 , . . . , mj ∈ { , n } i + j = 0 (mod 2) ∪ ( u i , v j ) : i ∈ { , m } j = 1 , . . . , n − i + j = 0 (mod 2) . Taking into account this equality, using Lemma 3.4, and denoting k ( a ν,µ ) k e ℓ mn = 1 mn X ν,µ ∈I mn | a ν,µ | , we get from (4.1) thatΛ LC mn ≤ max ( x,y ) ∈ [0 ,π ] k ( D mn ( x µ + x, y ν + y )) k e ℓ mn + O (cid:0) m − kD n k L ( T ) + n − kD m k L ( T ) + 1 (cid:1) = max ( x,y ) ∈ [0 ,π ] k ( D mn ( x µ + x, y ν + y )) k e ℓ mn + O (log n ) . (4.2)Next, we have D mn ( x, y ) = D mn ( x, y ) − d mn ( x, y ) , (4.3)where d mn ( x, y ) = X | k | /m + | l | /n =1 e i ( kx + ly ) = 2 m X k = − m e ikx cos (cid:16)n − nm | k | o(cid:17) . Repeating the proof of Lemma 3.3, it is not difficult to show that(4.4) sup y ∈ T k d mn ( · , y ) k L ( T ) . p log mp . Thus, using (4.3) and (4.4) as well as an analogue of (3.20) with the norm e ℓ mn instead of ℓ mn (to obtain the required equality, it suffices to repeat line by line the proof of (3.20)), we SYMPTOTICS OF THE LEBESGUE CONSTANT 15 derive k ( D mn ( x µ + x, y ν + y )) k e ℓ mn = k ( D mn ( x µ + x, y ν + y )) k e ℓ mn + O (cid:18) log n + p log mp (cid:19) = k ( S mn ( x µ + x, y ν + y )) k e ℓ mn + O (cid:18) log n + p log mp (cid:19) . (4.5)Next, by the same arguments as in the proof of Lemma 3.7, using (3.29) and (3.30), we get k ( S mn ( x µ + x,y ν + y )) k e ℓ mn = 2 m X B m (cid:12)(cid:12)(cid:12) ∆ (1) m ( x + x µ , nm ( y + y ν )) (cid:12)(cid:12)(cid:12) + (cid:16) log nm (cid:17) πm X B m (cid:12)(cid:12)(cid:12) ∆ (2) m ( x + x µ , nm ( y + y ν )) (cid:12)(cid:12)(cid:12) + O (log m )= 2 π (cid:12)(cid:12)(cid:12) sin mx + ny mx − ny (cid:12)(cid:12)(cid:12) (2 log m log n − log m ) + O (log n ) . (4.6)Now, combining (4.1), (4.5), and (4.6), we obtainΛ LC mn ≤ π (2 log m log n − log m ) + O (cid:18) log n + p log mp (cid:19) . (4.7)At the same time, it follows from (4.1) and (4.2) thatΛ LC mn ≥ k{D mn ( x µ +1 , y ν ) }k e ℓ mn + O (log n ) . Thus, repeating the above arguments with ( x, y ) = ( π/m, LC mn ≥ π (2 log m log n − log m ) + O (cid:18) log n + p log mp (cid:19) . (4.8)Finally, combining (4.7) and (4.8), we prove the theorem. (cid:3) Proof of Theorem 2.2.
By (2.2), we have L mn = π k D mn k L ([0 ,π ) ) . Using this equal-ity together with (3.17) and (3.31), we get (2.4). Similarly, applying (3.19) and (3.31), weprove (2.6). (cid:3) Proof of Theorem 2.3.
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