Growth of L p Lebesgue constants for convex polyhedra and other regions

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1 Growth of L Lebesgue constants for convex olyhedra and other regions J. Marshall Ash and Laura De Carli Abstract. For any convex olyhedron W in R m, (; ), and N, there are constants (W; ; m) and (W; ; m) such that N m( ) e (k x) N m( ) T m knw Similar results hold for more general regions. These results are various secial cases of the inequalities N m( ) e (k x) (N) ; T m kn where (N) N (m ) when ;, (N) N (m ) log N when, and (N) N m( ) when > where is a bounded subset of R m with non-emty interior.. Introduction.. One dimension. In one dimension there is a very well known estimate (.) e (kx) e (kx) T e (kx) 0 w 4 ln N, where e (x) e ix.[] This integral is called the Lebesgue constant. In higher dimensions there are as many Lebesgue constants as there are generalizations of an interval of integers, f N; N + ; ; Ng. Fix N and consider the oerator D dened by maing a Lebesgue integrable function f T! C to its N-th trigonometric artial sum s N (x) P ^f (k) e (kx). This oerator D is realized by convolving f with P e (kx). In short, D f! s N 000 Mathematics Subject Classication. Primary 45, 4A05; Secondary 408, 4A45. Key words and hrases. Lebesgue constant, Dirichlet kernel, kernels for convex sets, kernels for olyhedra. The rst author's research was artially suorted by a grant from the Faculty and Develoment Program of the College of Liberal Arts and Sciences, DePaul University.

2 J. MARSHALL ASH AND LAURA DE CARLI If we endow both the reimage and the image of f with the L T norm, then the oerator norm of D, kdk L!L, is dened to be su kfk ks N k. The Hausdor- Young inequality tells us that ks N k e (kx) f e (kx) kfk Using [ ;] in lace of f and letting! shows that P e (kx) is the oerator norm of D. Thus the oerator norm is equal to the Lebesgue constant, kdk L!L e (kx) This is the main reason for the imortance of nding a good estimate for the Lebesgue constant. If we consider the same oeration D, but consider D as a maing from f L to s N L,, then things roceed about the same as when was. Young's inequality gives ks N k e (kx) f e (kx) kfk ; where kk R jj T and again using [ ;] we arrive at a formula for the oerator norm of D, kdk L!L e (kx) ecause of this formula, the following lemma is also of interest. Lemma. 8 N < (.) j e (kx) j du N + 0 k0 where the constant R 0 sin u u O N 3 if > 3 O (log N) if 3 O () if < < 3 du; if k is a ositive even integer, then k k (.3) k (k )! k where the central Eulerian number may be dened to be k (.4) k ( ) k (k ) k 0 Most of this lemma aears elsewhere; one lace is in the reference [AAJRS]. Only the formula (.3) will be exlained below. ;

3 L LEESGUE CONSTANTS FOR POLYHEDRA 3.. Lebesgue constants for olyhedrons. If we let W [ ; ], then W R, an interval, may be thought of as a one dimensional convex olygon, and f N; N + ; ; Ng may be thought of as NW \ y y N W, a dilate of W by the scaling factor N. Let f L (T m ). An m-dimensional convex olyhedron W gives rise to a sequence of artial sums fs N g according to the formula s N (f) (x) ^f (k) e (k x) knw \ m where k m, x T m, and ^f (k) R f (x) e ( k x). Just as in the one T m dimensional situation, we again nd that for N xed and D dened by D f! s N we have for, (.5) kdk L!L e (kx) knw \ m Let D NW (x) knw \ m e (kx) There naturally occur two roblems good estimates for R T m jd NW (x)j and good estimates for R T m jd NW (x)j for >. The rst roblem has been very nicely solved by elinsky. Theorem. For any convex m-dimensional olyhedron W in R m and N, there are constants and such that (W ) log m ( + N) kd NW k jd NW (x)j (W ) log m ( + N) T m From now on we assume that > and focus on the question of getting good estimates for jd NW (x)j T m First note that if W [ ; ] m, then so (.6) D NW (x) knw \ m e (k x ) e (k x ) e (k m x m ) T m jd NW (x)j my j 0 This motivated the following theorem. 0 my N e (k j x j ) j k j N!m N e (kx) k N m m( ) N m( ) + o N j k j N m( N ) e (k j x j )

4 4 J. MARSHALL ASH AND LAURA DE CARLI Theorem. For any > and convex m-dimensional olyhedron W in R m and N, there are constants and such that (.7) kd NW k jd NW (x)j m( ) (W; m; ) N T m and (.8) (W; m; ) N m( ) kd NW k In short, N m( ) kdnw k N m( ) If the dimension is m, then inequality (.7) has already been established, rst for [YY], and then for all >.[A] The constants were more recise and the roofs were much more direct, but the roofs were number theoretic and dicult, and we do not know how to extend them to higher dimensions. Part of the argument given here will show that the result of [A] easily leads to the reverse inequality (.8) also when m. Another interesting roof for m and is in [NP]..3. Lebesgue constants for more general sets. We will also consider the more general case when W is allowed to have a more general shae, for examle, when W is relaced by a set R m that is bounded and has non-emty interior. When, the exected lower bound estimate of log m ( + N) is valid, but the corresonding uer value is larger. For examle, when is an m dimensional ball of radius N, then it turns out that (.9) T m e (k x) N m where A N means that AN is bounded above and below. Concerning the uer bounds in this generality, we have two results Theorem 3. For each m ;, and bounded set R m with non-emty interior, there is a constant such that the uer bound estimate holds, and T m jd N (x)j N m( ) Theorem 4. For each m ; (; ], and bounded set R m with nonemty interior, there is a constant such that the lower bound estimate holds. T m jd N (x)j N m( ) There remain two cases for general sets that we have not resolved () The question of an uer estimate for! kd N k e (k x) T m kn\ m

5 L LEESGUE CONSTANTS FOR POLYHEDRA 5 when (; ) and () the question of a lower estimate for the same integral when >. We can only rovide artial results in these two cases. Looking at the calculation (.6) above, in both cases we would like to get an estimate of the form cn m( ). In case (), this is not true. We calculate below that if is the m- dimensional unit ball so that N fx R m jxj Ng, then for ; and this choice of, we get the larger shar lower estimate of N m N m( )+, where m is ositive since ;. Interestingly, the exonent of N is constant as varies from to Theorem 5. 8 >< DNfjxjg > where m. N m N m( )+ for <, (m; ). N m( ) ln N for ; N m( ) for < To what extent this new candidate for an uer bound is actually extremal for all bounded sets with non-emty interior remains an oen question. In case (), we get the hoed for shar (since may be the cuboid [ ; ] m, see (.6) above) best lower estimate of cn m( ), but only by additionally assuming a small amount of regularity for the boundary of. For this, we get insiration from the aer of Liyand[L]. (See also [I], [Sh], and [Y].) Let A NW be the union of all cubes of side whose centers are at the oints of NW \ m. We say that W satises the M N condition (the M is for Minkowski) if ja NW 4 NW j (.0) lim N! N m 0; where the symmetric dierence oerator is dened by A 4 (A n ) [ ( n A). In other words, the measure of A NW 4 NW is o (N m ). If W is bounded with nonemty interior, then there are on the order of N m oints in NW, and so we can then exress that M N -condition by saying that the measure of A NW 4 NW must be \little oh" of the number of lattice oints in NW. Theorem 6. If the bounded set R m has non-emty interior and if satises the M N -condition, then for there holds the lower bound estimate for some constant (; m; ). N m( ) kd N k Another oen question is whether the boundary condition (.0) is necessary. The most otimistic set of conjectures might be that for bounded with nonemty interiors, N m( ) is the size of the lower estimate for all ; < <, and also the size of the uer estimate for ;, while N m( )+m( ) N m is the size of the uer estimate when is the size of the uer estimate when i extremal for ; ; and N m ln N. In other words, the shere is and the cuboid is extremal for >.

6 6 J. MARSHALL ASH AND LAURA DE CARLI Our intuition for this conjecture is that sheres and rectangles are extremal with resect to curvature, the former being everywhere curved and the latter nowhere curved. So it is at least lausible that for each value of one or the other should rovide the largest or smallest Lebesgue constant growth rate. In the L case, results for all convex sets in all dimensions are already known. See chater 6 of [L] for this. A small result of fairly great generality that does not seem to be well known is the M. Riesz Projection Theorem for a general half sace. This theorem is the obvious generalization of the very well known fact that the Riesz transform of an L (T m ) function is also in L. Theorem 7. Fix R m and b R. The associated hyerlane divides m into two halfsaces, L fx R m x dg H + fk m k dg and H fk m k < dg Then if f L (T m ), >, we have ^f (k) e (k x) kfk and kh + kh ^f (k) e (k x) kfk One might think that relacing fk k 0g by fk k dg would lead to a serious number-theoretic roblem. Fortunately, it does not. See the roof of Theorem 7 below. We will rove all results but one in the next section. The last section will consist of the roof of Theorem 5. In several of roofs we imlicitly assume that the origin is interior to the olyhedron W or the bounded set. The general case may be reduced to the case where the origin is interior to the olyhedron or bounded set by making an initial translation. Such a translation may introduce an error term of lower order, but will not aect the nal result. We will omit the details.. Most of the roofs We begin with a discussion of Lemma. As we already mentioned, everything excet the evaluation of k aears in reference [AAJRS]. To evaluate k, let N (x) be the characteristic function of [0; ) and N k N N N be the k-fold convolution of N. The function N m (x) is called the mth order cardinal -sline. Then N d k, k (x) cn (x) so setting x k in the double inversion relation N k (x) dnk (!) _ (x) shows that N k (k) sin x x k 0 sin x x whereas on age 9 of [C] a direct comutation shows N k (k) to be equal to the sum (.4) divided by (k )!. Finally, the identication of the sum (.4) as the central Eulerian number aears in reference [GKP]. k ;

7 L LEESGUE CONSTANTS FOR POLYHEDRA 7 Remark. The left hand side of relation (.) can also be evaluated exactly by exanding and using orthogonality when k and this leads to the olynomial relation k (N )k 0 a Nk () where the a Nk () are dened by k (k )! N k + O N + x + x + + x N (N )k k 0 This is immediate from the above result and the calculation N 0 e (x)! k N 0 e ( x)! k (N )k 0 (N )k 0 k 3 a Nk () x (N )k a Nk () e (x) 0 a Nk () + j60 C j e (jx) a Nk () e ( Actually it aear that R 0 P N 0 e (x) k is in general an odd olynomial in N of degree k. Relations (.) and (.3) rovide a formula for the leading coecient of this olynomial. It might be interesting to nd formulas for the other coecients. We turn to establishing the uer bound inequality (.7). If W is a convex m-dimensional olyhedron in R m and N, then W is a bounded set with a non-emty interior. A bounded subset R m with non-emty interior has two geometric roerties. Namely, there are constants d () and D () so that (.) dn m jn \ m j ; and (.) N D [ N; N] m [ DN; DN] m We will use three lemmas, all of which are essentially from the roof of Theorem on ages 404 and 405 of [T]. For x T m, and f L (T m ), we dene the m-dimensional de la Vallee-Poussin mean to be 0 (.3) N (f) N + jk j j A ^f (k) e (k x) ; SP (M) N<jk jjn;js jk Y js N where P (M) is the ower set of M f; ; ; mg. The sectrum of a trigonometric olynomial P c k e (k x) is the set of k m for which c k 6 0. Lemma. Let N (f) be the m-dimensional de la Vallee-Poussin mean of f L (T m ) and let W satisfy conditions (.) and (.). Then () The cardinality of the sectrum of DN (f) is O (N m ), x)

8 8 J. MARSHALL ASH AND LAURA DE CARLI () (.4) s NW (f) s NW ( DN (f)) ; and (3) for all, (.5) k DN (f)k 3 m kfk Proof. To make sense of DN, D must be an integer, so if necessary simly relace D by dde. Then () is immediate from the denition of DN. For (), note from the denition of that when all jk j j DN, S? and the multilier of ^f (k) e (k x) is. We now establish (3). For f L (T), the de la Vallee-Poussin mean n (f) is dened to be P ^f jkjn (k) e (kx)+ P n<jkjn From this we see that (.6) n (f) n + n + n n (f) n + n (f) ; n+ jkj n+ ^f (k) e (kx). where the Cesaro mean n (f) corresonds to convolution with a ositive kernel k n. R For, k n (f)k kk n k kfk 0 k n kfk kfk. From this and (.6) it follows that (.7) k n (f)k 3 kfk Dene j DN (f); f L (Tm ) to be the de la Vallee-Poussin mean of f thought of as a function of x j with the other m coordinates of x xed. We nd that DN (f) DN m ( m DN ( ( DN (f)) )). This and inequality (.7) lead to (3). Lemma 3. Let (0; ] and let T (x) ; x T m, be a trigonometric olynomial with sectrum of cardinality n. Then for q >, kt k q (m; ; q) n q kt k This is roved on age 3 of [T]. We refer to it as the change of norm lemma. It is a sort of reverse Holder's inequality since kt k kt k q is almost immediate from Holder's inequality. Lemma 4. Let K be an arbitrary olyhedron in R m. Then there is a constant deending only on K, such that for all n N and for all (; ) there holds m ks NK (f)k + 4 kfk This is art of Corollary.4.5 on age 56 of [T]. This is roved carefully, but we would like to ll in a detail at one oint. The rst ste of the roof is to show that if f L (T m ) and if H is a half sace of m having a hyerlane through the origin as its boundary(h fk m k 0g for some R m ) or as the boundary of its comlement(h fk m k < 0g for some R m ), then f (k) e (k x) kh kfk This is Theorem.4.3, the M. Riesz Theorem on Projections, on age 54 of [T]. The second ste is to assert that the same result holds if the bounding hyerlane

9 L LEESGUE CONSTANTS FOR POLYHEDRA 9 does not ass through the origin. The third ste is to decomose a general olyhedron into convex olyhedra of m + sides. The last ste is to intersect half saces in a clever way in order to get the result for a convex olyhedron of m + sides. Stes, 3, and, 4 are carried out carefully on ages 54{57 of [T]. We will now do ste by roving Theorem 7. Proof. We are given f L (T m ), >, xed R m, and xed b R. We must show ^f (k) e (k x) kfk and ^f (k) e (k x) kfk ; kh + where the hyerlane L fx R m x dg divides m into the two halfsaces, H + fk m k dg and H fk m k < dg We have just mentioned that this has already been roved in [T] when H + and H are relaced by H 0 + fk m k 0g and H 0 fk m k < 0g. The idea of this roof is to translate the general case of H + into the secial case of H 0 +. We will only rove that for f L (T m ) ;, there holds the inequality (.8) f + kfk ; where f + (x) P ^f kh (k) e (k x). This result was already roved in [T] when + H + is relaced by H 0 +. The standard interolation argument aearing in [T] will then roduce the rst art of our conclusion; also the case of H can be treated in a very similar way. Without loss of generality we may assume jj. Reversing if necessary, we may also assume that d > 0. First assume that f has only a nite sectrum, i.e., that f is a trigonometric olynomial. Move the halfsace H 0 + in the direction of a distance of d to make it coincide with H +. Then move it a little further if no oint of the sectrum of f lies on the boundary hyerlane until its edge rst touches a oint which is in the sectrum of f. Let ` be any such oint. Observe that this maneuver might not have been ossible if the sectrum of f had been innite since there could have been no sectrum oints on the boundary hyerlane, but an innite sequence of sectrum oints aroaching that hyerlane. Let g (x) e ( ` x) f (x). Since jgj jfj, g L, so by the result in [T], P m0 ^g (m) e (m x) has controlled L norm, P m0 ^g (m) e (m x) kgk. ut letting m k `, this kh last sum becomes e ( ` x) P k` ^g (k `) e (k x) e ( ` x) f + (x). Finally, je ( ` x)j and kf + (x)k kfk follows. ut the inequality constant is indeendent of the sectrum size and the trigonometric olynomials are dense in L, so the general case of innite sectrum follows by aroximating f by a sequence of trigonometric olynomials. We give the details. Dene the square (C; ) means to be this trigonometric olynomial where n (f) s ii m (f) (n + ) m k i n i 0 i m n s ii m (f) ; i m0 ^f (k) e (k x) i k m i m

10 0 J. MARSHALL ASH AND LAURA DE CARLI Since, in articular f L, so from P kc ^f (k) P ^f (k), it follows that f + L. It is well known [, Vol. II, Theorem 3.8] that since f + L, (.9) n f +! f + a.e. as n!. Alying the nite sectrum inequality (.8) to n (f), we get n f + ( n (f)) + k n (f)k kfk ; where the last ste was exlained in the roof of art (3) of Lemma above. Now let n!, using (.9), the last inequality, and Fatou's lemma to get inequality (.8) without any constraint on the sectrum of f. The case of H is exactly the same. We may now ass to the roof of inequality (.7). Proof. Let f L (T m ) and let W be a convex olyhedron. From equation (.5), to rove inequality (.7), it suces to nd a constant C(m; ; W ) so that ks NW (f)k CN m( ) kfk. Pick an integer D so large that condition (.) holds for W. Aly rst (.4), then Lemma 4, then Lemma 3, and nally (.5) ks NW (f)k ks NW ( DN (f))k m + 4 k DN (f)k m C + 4 N m( ) k DN (f)k m 3 m C + 4 N m( ) kfk (W; m; ) N m( ) kfk The following simle lemma allows us to convert an uer bound estimate for an L 0 norm of a Dirichlet kernel into a lower bound estimate of the L norm of that kernel. Lemma 5. If S is any nite subset of m of cardinality n, jsj n, and if D D (S) e (k x) ; ks then (.0) kdk n kdk 0 where and and 0 are conjugate exonents, + 0.

11 L LEESGUE CONSTANTS FOR POLYHEDRA Proof. Aly Plancherel's equation to get n e (k x) jddj ks T m T m ks and then Holder's inequality 0 jddj jdj jdj 0 kdk kdk 0 T m T m T m We can now give a quick roof of the lower bound inequality (.8). Proof. For every 0 in (; ) we have kd NW k 0 (W; m; ) N m( 0 ) From inequality (.), Lemma 5, and this, we get kd (NW \ m dn m )k kd (NW \ m )k 0 dn m (W; m; ) N m( 0 ) d (W; m; ) N m 0 (W; m; ) N m( ) ; where is dened to be d. This establishes the lower bound inequality (.8). This comletes our work on olyhedra. We move on to the roof of Theorem 3. The only geometric roerties of bounded sets with non-emty interiors that we will need will be inequalities (.) and (.). Proof. Since Young's inequality gives + + (.) kd (N \ m ) T DN (f)k kd (N \ m )k kt DN fk ; + where the requirement that + follows from. Since T DN (f) has at most O (N m ) terms, we have the change of norm inequality (.) kt DN (f)k + ; C kt DN (f)k (N m ) C (N m ) We also use roerty (.5) of the delayed means (.3) kt DN (f)k 3 m kfk The nal item is Plancherel's Theorem, (.4) kd (N \ m )k kn\ m + kt DN (f)k! CN m

12 J. MARSHALL ASH AND LAURA DE CARLI Putting together inequalities (.) to (.4) gives kd (N \ m ) fk kd (N \ m )k kt DN (f)k + kd (N \ m )k C (N m ) CN m 3 m N m( N m( ) kfk ) kfk kt DN (f)k Theorem 4 is immediate from this and Lemma 5 Proof. We are assuming that (; ] so that 0 [; ) and that is bounded with non-emty interior. We simly reeat the roof of inequality (.8) given above, excet that 0 is constrained to [; ) and W is relaced by. We next rove Theorem 6. Proof. We are assuming that is bounded, has non-emty interior and satises the M N -condition. We need to show that for every, the L (T m ) norm of D N (x) is bounded below by N m( ). We take insiration from the aer of Liyand [L]. Let A N be a union of cubes of side whose center is at oints of N \ m. Let Q(z) be the cube of side centered at z. The Fourier transform of the characteristic function of Q(0), the cube of side centered at the origin, is my e ( x z) dz e ixz sin(x j ) dz, and so the Fourier transform Q(0) Q(0) x j j my sin(x j ) of the characteristic function of Q(z) z + Q(0) is e ( z x). Then, x j j e ( u x) du e ( u x) du A N Q(z) my j sin(x j ) x j zn\ m zn\ m e ( z x) my j sin(x j ) D N (x); x j and we have roved the following imortant identity. my sin(x j ) (.5) e ( u x) du D N (x) A N x j j y the elementary inequality sin x x see from jaj jaj that (.6) kd N k e ( u x) du T m A N which is valid when jxj, we can ^ AN Integrating e ( u x) over A N and noting that A N N (NA N ) [ (A N N) gives ^ AN (x) ^ N (x) ^ NAN (x) + ^ AN N (x)

13 L LEESGUE CONSTANTS FOR POLYHEDRA 3 Take the L (T m ) norm of e ( u x) and aly the triangle inequality to get (.7) ^ AN k^ N k ^NAN ^AN nn Note that we reserve kk for kk L (T m ) ; however, we will always write kk L (R m ) without abbreviation. Since T m R m, we have L L ^ NAN ^ NAN ^ (T m NAN ; ) (R m ) and since, from the Hausdor-Young inequality there follows L ^ L NAN (R m ) NAN jn n A Nj ; 0 (R m ) so that (.8) ^ NAN jn n A N j The same reasoning also roduces (.9) ja N n Nj ^ AN N A change of variables yields k^ N k j^ N (x)j N m( ) T m N m( ) k^ k L (R m ) k^ k L (R m nnt m ) NT m j^ (x)j Combine this with relations (.6), (.7), (.8), and (.9) to get 0 k^ k L (R m ) k^ k L (R m nnt m ) m( ) (.0) kd N k N jnna N j +ja N nnj A N m( ) The Hausdor-Young inequality imlies that ^ L (R m ). (In fact, k^ k L (R m ) jj.) So by Lebesgue's dominated convergence Theorem, k^ k L (R m nnt m ) tends to 0 as N!. ecause of the M N condition, both jn n A N j and ja N n Nj are o (N m ), so the term in curly brackets also goes to zero when N!. Thus from inequality (.0) it follows that there is a constant so that (.) jd N (x)j N m( ) T m Remark. This roof showed that k^ N k is asymtotic to cn m( ) where c k^ k L (R m ). Since inequality (.6) can be reversed (with constant ()m instead of ), an uer bound for the case of L requires no extra work. However, this is not useful since the case for an uer bound when was already done without the boundary assumtion M N in the roof of Theorem 3.

14 4 J. MARSHALL ASH AND LAURA DE CARLI 3. Proof of Theorem 5 We will only estimate the value of kd N k where fx R m jxj g from below. More secically, we will show that kd N k ( N m < N m ln N We will rove below this lemma. h Lemma 6. For every ; there is a ositive constant c (; m) so that for all R >, (3.) (3.) If R e ( v y) dv dy c R +m m +, there is a ositive constant c (m) so that for all R >, R e ( v y) dv dy c ln R h Let ; and assume temorarily the validity of inequality (3.). For a small ositive < (to be chosen later) we have kd N k jd N j jd N j ; T m The same reasoning that led from identity (.5) to inequalities (.6) and (.7) above roduces here jd N j Concatenating the last two inequalities, [N (x) j N \ AN (x)j j AN \ N(x)j (3.3) kd N k ( [N (x) j AN \ N(x)j + j N \ AN (x)j ) Making rst the change of variable u Nv; du N m dv and then the substitution y Nx; dy N m, we estimate the rst term on the right hand side of (3.3)

15 L LEESGUE CONSTANTS FOR POLYHEDRA 5 [ N (x) N e ( N m N m( ) u x) du e ( v Nx) dv e ( v y) dv N N m( ) c +m (N) dy (3.4) c N m m( ) The last inequality follows from the estimate (3.). Since < we may estimate each of the last two terms in (3.3) by alying Holder's inequality with exonents and ( ), then exanding the domain of integration to R n, and then alying the Plancherel equation. We get j AN \ N(x)j j AN \ N(x)j (vm m ) and ja N Nj v ( j N \ AN (x)j jn AN j v ( m m ) m( ) ) m( where v m! m m m (m (m)) is the volume of the m dimensional ball. ecause of the geometry of the m-shere, we know that (3.5) ja N Nj + jn A N j O N m [S] Thus there is a constant c (; m) so that the quantity in curly brackets in inequality (3.3) is c m( m ) N. From (3.3), this, and inequality (3.4) we get ) kd N k c N m m( ) c m( m ) N m( ) c c N m N m We now choose. Choose it so small that > 0. If, we do not need the scaling trick. Let. Then inequality m + (3.3) becomes kd N k [ N (x) ( ) j AN \ N(x)j + j N \ AN (x)j ;

16 6 J. MARSHALL ASH AND LAURA DE CARLI so that (3.6a) kd N k [N (x) c 0 N m and using estimate (3.) instead of estimate (3.) allows us to relace inequality (3.4) with [N (x) e ( u x) du N c N m ln N Putting this together with (3.6a) gives (3.). It only remains to rove the lemma. Proof. It is well known that c (x) e ( x ) J m ( jxj) jxj jj m ; where J is the standard essel function of order. (See Stein & Weiss,.70, Thm. 4 with 0). So we have to estimate j c (x)j e ( v y) dv dy R R 0 R! m Jm (r) r m r m dr Since J m (r) is asymtotic to O r m for small r; the integrand is bounded for small r so a lower estimate of the form cr with > 0 will hold equally well for this integral or for I () m m R a J m (r) r m+m dr; where a is any xed ositive constant. In other words, to rove the lemma it suces to nd a ositive c (; m) for which there holds ( c (3.7) I R +m if < c ln R if In the integral dening I, substitute s r to write (3.8) I R a Jm (s) s m m ds A well known large variable estimate for essel functions is that r cos s 4 J m (s) + O [E] s s 3 It follows from this that (3.9) I where J R a r J + E; cos s 4 s m m ds s

17 L LEESGUE CONSTANTS FOR POLYHEDRA 7 and E is on the order of R (3.0) a s 3 s m m ds O R m o R m We will underestimate J by relacing cos s 4 by and by 0 elsewhere. Secically, if than or equal to k a + m ; R + m + 8 [b; t] 6 when it is greater where de denotes the least integer and bc denotes the greatest integer, then m + I k k 4 3 ; k m + + [a; R] 4 3 and for s I k ; cos s 4. Hence t J s sm m ds where its smallest value to get J t kb kb I k t kb I k s ds so that >. Underestimate each integrand by k m + + sgn ( ) Finally note that b is bounded, t R c where c is bounded, and use the integral test estimate of comaring P f (k) with R f (k) for ositive monotone f to see that there is a ositive constant d (; m) so that ( d J R R +m if < d ln R if Putting this estimate and estimate (3.0) into equation (3.9) allows us to comlete estimate (3.7) and thereby nish the roof. References [A] J. M. Ash, Triangular Dirichlet kernels and growth of L Lebesgue constants, rerint. [AAJRS]. Anderson, J. M. Ash, R. Jones, D. G. Rider,. Saari, Exonential sums with coecients 0 or and concentrated L norms, to aear in Annales de l'institut Fourier. [C] C. K. Chui, An introduction to wavelets. Wavelet Analysis and its Alications,. Academic Press, Inc., oston, MA, 99. [E] A. Erdelyi et al., Higher transcendental functions. Vol. II. ased, in art, on notes left by Harry ateman. McGraw-Hill, New York-Toronto-London, 953. Page 85, 7.3.(3). [GKP] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics. A foundation for comuter science. Second edition. Addison-Wesley Publishing Comany, Reading, MA, 994. [I] V. A. Ilyin, Problems of localization and convergence for Fourier series in fundamental systems of the Lalace oerator, Usekhi Mat. Nauk 3(968), 6{0. (Translation in Russian Mathematical Surveys 3, 59{6.)

18 8 J. MARSHALL ASH AND LAURA DE CARLI [K] [L] [NP] [N] [Sh] [S] [SW] [T] [Y] [YY] [] A. Ya. Khinchin, Continued Fractions, Nauka, Moscow (978). Also translated from the third (96) Russian edition. Rerint of the 964 translation. Dover Publications, Inc., Mineola, NY, 997. E. R. Liyand, Lebesgue Constants of multile Fourier series, Online Journal of Analytic Combinatorics, Issue (006), # 5, -. F. L. Nazarov and A. N. Podkorytov, On the behavior of the Lebesgue constants for two-dimensional Fourier sums over olygons, St. Petersburg Math. J., 7(996), 663{680. (Translation age numbers are 663{680.) Niven and uckerman, An Introduction to the Theory of Numbers, 3rd edition, Wiley, 97. H. S. Shairo, Lebesgue constants for sherical artial sums, J. of Aroximation Theory, 3(975), 40{44. E M. Stein, Harmonic analysis real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murhy. Princeton Mathematical Series, 43. Monograhs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 993. P. 36{363. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Saces, Princeton Mathematical Series, No. 3. Princeton Univ. Press, Princeton, N.J., 97. R. M. Trigub and E. S. elinsky, Fourier Analysis and Aroximation of Functions, Kluwer Academic Publishers, Dordrecht, 004. V. A. Yudin, ehavior of Lebesgue constants, Translated from Matematicheskie ametki, 7(975), 40{405. (Translation is in Mathematical Notes 7 on ages 33{35.) A. A. Yudin and V. A. Yudin, Polygonal Dirichlet kernels and growth of Lebesgue constants, Translated from Matematicheskie ametki, 37(985), 0{36. (Translation is in Mathematical Notes 37 on ages 4{35.) A. ygmund, Trigonometric Series, v., nd ed., Cambridge University Press, New York 959 Mathematics Deartment, DePaul University, Chicago, IL address mash@math.deaul.edu URL htt// Current address Florida International University, Deartment of Mathematics, University Park, Miami, FL address decarlil@fiu.edu URL htt//