Fourier Expansions of Functions with Bounded Variation of Several Variables

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Fourier Expansions of Functions with Bounded Variation of Several VariablesAuthor(s): Leonardo ColzaniSource: Transactions of the American Mathematical Society, Vol. 358, No. 12 (Dec., 2006), pp.5501-5521

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transactions of theamerican mathematical societyVolume 358, Number 12, December 2006, Pages 5501-5521S 0002-9947(06)03910-9Article electronically published on July 20, 2006

FOURIER EXPANSIONS OF FUNCTIONS WITH BOUNDEDVARIATION OF SEVERAL VARIABLESLEONARDO COLZANI

Abstract. In the first part of the paper we establish the pointwise convergence as t ? *+oo for convolution operators fRd tdK (ty) (p(x ? y)dy under theassumptions that ip(y) has integrable derivatives up to an order a and that|^(y)| < c (1+ 13/1)with a + ? > d.We also estimatetheHausdorffdimension of the set where divergence may occur. In particular, when the kernel is theFourier transform of a bounded set in the plane, we recover a two-dimensionalanalog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove anequiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows usto transfer some of the results proved for Fourier integrals to eigenfunctionexpansions. Finally, we present some examples of different behaviors betweenFourier integrals, Fourier series and spherical harmonic expansions.

Fourier integrals on Euclidean spacesThis section is devoted to the study of the pointwise convergence as t?> +00 foroperators of the type

X (i_1C) exp(27ri?r)de/ tdK ty) p(x y)dy.JRd JRdHere (?) exp(27n?x)d? ip(x).^+?? JtnA classical reference forthis problem is [3].Other referencesmore directly relatedto thispaper are [1], [4], [5], [6], [15], [16], [19], [23], [24], 25]whichare devotedto Fourier expansions of piecewise smooth functions, [2] and [22],which estimatethe capacity of the divergence sets of one-dimensional Fourier series, [7], [12], [17],[26],which contain results on spherical summability of Fourier integrals of functionsin Sobolev classes, [18],with a simple proof of the almost everywhere convergenceof expansions in eigenfunctions of functions in Sobolev classes. Finally, a specialmention should be made to the research tutorial [21]. In order tomotivate what

t

Received by the editors April 26, 2004 and, in revised form, November 16, 2004.2000 Mathematics Subject Classification. Primary 42B08, 43A50.?2006 American Mathematical SocietyReverts to public domain 28 years from publication

5501

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5502 LEONARDO COLZANI

follows, let us consider two extreme cases. If the function d, then |^(?)| < c(l + |?|)~a is integrable. Hence,ifx (0 is bounded and continuous at the origin, thenlim/ x(^10^(Oexp(27T^K = X(0)^(x).Ifthe kernelK(x) has thedecay \K(x)\ c(l + \x\)~? ith? > d, thenK(x)is integrable. Hence, if (p(y) is bounded and continuous at the point x, then

lim/ tdK(ty)tp(x-y)dy d are sufficient conditionsfor the pointwise inversion of the Fourier transform, with a possible exception ofa set of points with Hausdorff dimension at most d ? a. In what follows thesmoothness of a function ismeasured with theRiesz potentials or fractional powersAa/2 of the Laplace operator A = ? V d2/dx2, defined spectrally by Aa/2(p(?) =(2-ir\?\)a 0,/JrX (?_1?) (0 exp(27r??x)dC


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FUNCTIONS WITH BOUNDED VARIATION 5503

holds with thepossible exception of a set ofpoints withHausdorff dimension atmostd ? a. More precisely, assume that at a point x there exists 0 < e < 1 such thatwhen r ? 0+,

[ Aa/2ip(x-y) dya + ?-d.j tdK(ty) d is bestpossible. By substituting the square root of the Laplace operator with the gradient,one can apply the theorem to functions with bounded variation, that is, integrablefunctions whose distributional firstderivatives are finitemeasures. In particular,the theorems apply to piecewise smooth functions, which are linear combinationsof functions x?(x)x?(x), products of smooth functions and characteristic functionsof bounded domains with smooth boundary. The content of the following theoremis that in this case convergence holds everywhere, even along the discontinuities of

the functions expanded.Theorem 3. Let %(?) be an integrable Fourier multiplier, with compact support,smooth in a neighborhood of the origin and with x(0) = 1. Assume that the associatedconvolution ernelK(y) has decay \K(y)\ c(l + \y\)~forsome? > d? 1.If (f(y) is a piecewise smooth function suitably normalized along the discontinuities,then at every point x,lim/ x(?"10^(Oexp(27ri^K= lim/ tdK ty)


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integrals of functions with bounded variation, and in dimension two, if the domainhas just a bit of smoothness, one obtains a generalization of thisDirichlet theorem.Corollary 5. Let ?? be a bounded planar domain containing the origin and assumethat \xn(y)\ c(l + |y|) for some? > 1. If ip(y) s an integrableunctionwithbounded variation, then,with a possible exception of a set ofpoints with Hausdorffdimension at most 1,

lim / ip(x).Moreover, if ip(y) is piecewise smooth, then convergence holds at every point.

The above corollaries are essentially already known and they are included hereonly because they easily follow from our main results and have been the originalmotivation forour work. In particular, Corollary 4 fora restricted range of indexesis already contained in [12] and [26],while Corollary 5 for spherical summabilityis already contained in [21], Sections 3, 4, 5, and Propositions 4 and 8. Generalpartial sums of Fourier integrals of piecewise smooth functions are discussed in [1]and [6] nd the associated Gibbs phenomenon n [4], [8], [9], [10], [11], nd [25].An example of domain which does not satisfy the assumptions of Corollary 5 is adomain with flat boundary, since Xto(v) nas a decay (1+ |y|)~ ify ?> oo alongdirections perpendicular to flat parts of the boundary. In this case convergenceof Fourier expansion may fail. Since the uniform decay of Fourier transforms ofcharacteristic functions is never better than c (1 -h |2/|)_ , the corollary doesnot apply to functions with bounded variation in dimensions greater than two.Indeed, in dimension d > 3, the partial sums of Fourier integrals of piecewisesmooth function may not converge, even at points where the functions expandedare smooth; see [20]and [16] or the examples in the last section of the paper. In thesecond section of the paper we shall give an extension ofCorollary 4 to expansionsin eigenfunctions of elliptic operators, for a restricted range of indexes, and in thefinal section we shall present some examples of different behaviors between Fourierintegrals, Fourier series and spherical harmonic expansions.

Proof of Theorem 1. If x(?) is smooth around the origin, there exists a smoothcutoff i?(?) identically one in a neighborhood of the origin and zero in a slightlylargerneighborhood such that VK0x(0 is smooth everywhere. Denoting by tdK(ty)? tdA(ty) + tdB(ty) the convolution kernels associated to the dilated multipliersX (t'1^) = ip i_1?)x (?-1?) + (1- ^ (?-1?)) X (?-1?)> one obtains thedecomposition/ x(*-1i)^(Oexp(27riex)d?= tdK(ty)


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/Rd (y)dy= *l>0)x (0)= 1.Hence,

\[ tdA(ty)ip(x-y)dy-y(x)= / idA(ty)


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5506 LEONARDO COLZANI\OLd\y\a . If i?j(y) is a smooth cutoff identically one in {\y\ 1} and identically zeroin{\y\ 2}, then

/ \yrd?(x-y)dy=[ (l-^(y))\yrdv(x-y)dyJRd JRd+ / ^(y)\y\a v(x-y)dy.JRd

\a?dThe kernel (1 ? ip(y))\y\ ~ is smooth and, by the assumptions on /?(y), thesecond integral is absolutely convergent and defines a smooth function. It thensuffices to prove that x is a Lebesgue point of the third convolution,/ / V>(y) y\a?(x-y- z)dy l i?(y)\y\a/?(x y)dy

J{\z\(y-z)\y-z\ -il>(y)\y\a dz \Kx-y)\dy\y\ad 1\v(x-y)\dy.crd [ \yrd\fi(x-y)\dycrd^ [J{\y\


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Assume that L ,


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This estimate suggests studying the set of divergence in terms of capacity. Theapproach followed in the above proofs of Theorem 1 and Theorem 2 is perhapsa bit more complicated, however, it has the advantage of giving a quite explicitdescription of the set of points where convergence takes place and an estimate ofthe speed of convergence.Proof of Theorem 3. Let ip(y) be piecewise smooth and let Vip(y)dy ? ?i(y)dy +v(y)do(y) be the decomposition of its gradient into an absolutely continuous part/J>(y)dynd a singular part v(y)da(y) supported on a smooth surface S. Then, asin the proof of Theorem 1,

/ tdK(ty)ip(x-y)dyJRd= / tdA(ty)ip(x y)dy f VA"1 (tdB(ty))V


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Since \C(z)\ < c(l + \z\) is integrable with respect to a (d ? l)-dimensionalsurface measure, given e > 0, there exists R > 0 such that

/ {tx-tS}n{\z\>R} C(z)da(z)


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t>d

other second order elliptic differential operators with appropriate boundary conditions. An integrable function on M.dwith support onV can be expanded inFourierintegralswith respect to the trigonometric system {exp(27r^x)} and also inFourierseries with respect to the orthonormal system {tp\(x)},

/ 7ri?x)d?


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Observe that the range of indexes 7 > d ? a ? 1 forRiesz summability ofeigenfunction expansions onmanifolds is smaller than the range 7 > (d? l)/2 ? aforRiesz summability ofmultiple Fourier integrals in Corollary 4. Also, observethat in the above theorem Aa/2(p(y) is required to be an integrable function, notjust a measure as in the previous section. We will see that these restrictions arenecessary. The following proof is essentially taken from [4].Proof of Theorem 8. For simplicity we write x(u) instead of(l ? t~2u2y ,droppingany explicit reference on 7 and t. Indeed, inwhat follows, the precise form of x(u)does not play any particular role. Let p(u) be an even test function on ? 00 e, then

(X Kl)-

C *X (ICI)) fi? exp(27r??x)deV C*X(A), a)a(x).Rd AProof of Lemma 9. By the spectral decomposition of the Laplace operator,

/ C*x(l?l)?(?)exp(27ri?x)d?= ((s)x(s)cos sy/?^)


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Lemma 10. //Aa/2(p(y) is an integrable function and a + 7 > d ? 1, then

lim/ |x(l?l)-C*x(KI)ll?(O|de0,t^+00 jRdlimVJ|x(A)-C*x(A)||^,Va)||V'a^)| = 0.AProof of Lemma 10. The convolution ? * x(t?) is a good approximation of x(u)away from the singularities of this function. More precisely, ? (u) has integral oneand other moments zero and it is concentrated in a neighborhood of the origin;moreover, in a neighborhood of the singularities ?t the function x (u) has size ?~7.This implies that \x u) ? C *X (u)\ is essentially a sum of two bumps of height t~Jconcentrated around ?t,

\x(u)-?*x(u)\


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Now observe that, by the relation (Aa/V^A) = (27rA)a(V,^A) and the orthonormality of the system {ip\(x)},1/2E i|A-s|


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can be substituted by more general operators. In particular, ifx(u) ? min{l, ua},then the operator associated to ( * x (u) has a kernel concentrated around thediagonal {\x

?y\ e}, while the kernel associated tox (u)~C*X iu) isbounded, thisisbecause ofthe fastdecay \x(u) C*x(u)\ < c (1+ M )~~V-his provesthattheoperator Aa/2 is pseudolocal. Although the above result for general eigenfunction

expansions is essentially sharp, following a suggestion by M. Taylor we give animprovement for classical multiple Fourier series on the torus. This is related to theGauss problem of estimating the number of points with integer coordinates in largespheres. The torus Td = Rd/Zd can be identified with the unit cube [-1/2, +l/2)dand an integrable function on Rd with support on Td can be expanded both inFourier integrals and Fourier series,

/ (p(?) xp(2m?x)d?= d follows from the integrability of the Fourier transform of the functionexpanded, while the result for7 > (d ? l)/2 follows from the integrability of theRiesz kernels. Hence in the sequel we may assume a < d and 7 < (d ? l)/2. Leta(y) be a smoothfunction ith a(y) = 1 if \y\ 2 and a(y) = 0 if \y\ 3 anddefine

*(y)= X>(*~lfcMfc + 2/)kezdThis $(y) is sum of a finite number of non overlapping copies of ip(y) and thesetwo functions agree inTd. As in the proof of Theorem 8, let x(^) = (l ? ?_2,l?2) >let p(u) be an even test function with mean one and other moments zero and with



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Fourier transform p(s) supported in {\s\< 1}, also let ?(u) = tp(tu). Then

/ x(l^l)^)exp(27r^K- Y,x(\k\)(p(k)exp(2mkx)Rd kezd1/x(l l)^(?)exp(27r


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the annulus {||?| ? t\ t-1} contains no more than c^-2+2/(d+1) points with integer coordinates. This is a consequence of good estimates for the number of integerpoints inthespheres |?|


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Finally, observe that the restriction a + 27 > d ? 1 in Lemma 14 ismore demanding than the restriction a + 2j > d ? 2+ 2/(d + 1) in Lemma 13. Defining((u)

=t1~?p (tl~?u) and $(y)

=Y,kezd a (??_1*) (* y) with a suitable 5 > 0,one obtains a worse Lemma 13 but a better Lemma 14 and at the end a bettertheorem. This concludes the proof ofTheorem 11. D

Examples, counterexamples, concluding remarksIn this final section we want to compare the results obtained on euclidean spacesand on manifolds. Let us first consider the euclidean space Rd. Corollary 4 guarantees that Riesz means with index 7 > (d ? l)/2 ? a of functions with a integrable derivatives converge at all points where the functions are smooth. The

following example shows that indeed the result is essentially sharp. The function(p(y) = (1 ? \y\+_1 has a bit less than a integrable derivatives and its Fouriertransform has asymptotic expansion

Tr^rXa) |?p-(d-2>/2 Ja+{d_2)/2(2n d)? 7T-aT(a) l^p-^-D/2 cos (2tt |?|- (2a + d - 1)tt/4) .

Hence the Riesz means of this function at the origin x = 0 have asymptoticexpansion

Tr^IXa) / (l-li-1^2)7 ira-(d~2)/2Ja+(d-2)/2(27r|?|KRd V J^ ^(d+D/2-a f1 (j _ r2y r(d-i)/2-a cos(2?rtr - c)dr.Jo

The singularity of r(d-1)/2-?i when r ?> 0+ gives to the oscillatory integral adecay ?a-(d+i)/2^ which compensates the factor in front of the integral, while thesingularity of (l ? r2)7 when r ?> 1? gives a decay t~1~1 with an oscillation.Hence, themeans with index 7 < (d? l)/2 ? a do not converge at the origin, whichis a point where the function is smooth. This phenomenon isdue to the singularitiesof the function expanded that, propagating from the sphere {\y\ 1}, focus at theorigin. This example also exhibits the speed of convergence in Theorem 2.

We now turn to the torus Td. In Theorem 11we proved the Riesz summabilityofmultiple Fourier series with 7 > (d? a ? 1? e) ?2. The following example forthe spherical summability 7 = 0 shows that the best possible e is indeed small.Let ? (y) = Ylkezd exP (27rifcy)be themultiple Fourier series associated to the unitmass inTd concentrated at zero and definetpy) A-a/2? y) E (27r)~a exP 2 ky)kezd-{o}

This function on the torus is smooth, except for a singularity at the origin oftyPe \y\a~ Now observe that at the point x = (1/2,1/2,1/2,...) all exponentialsexp(2nikx) on the spheres {|fe| = t} take the same value, +1 if |fc| = t2 is evenand ? 1 if |fe| = t2 is odd. Hence, when grouped spherically the terms of theFourier series have size (27rt)~a \{\k\ t}\. Since the number of integer points onthe spheres {|fe| =t} can be of the order of td~2, for the convergence of the Fourierseries one has to require at least a > d? 2. Recalling that spherical summability ofFourier integrals holds ifoj > (d ? l)/2, one deduces that the range of indexes forspherical summability ofFourier series ismore restricted than the one for integrals.



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Finally, in Theorem 8 we proved that on a compact manifold of dimension dRiesz summability holds provided that 7 > d ? a ? 1. Despite the fact that thisindex is greater than the previous ones forRd and Td, the result is sharp. Thefollowing example for spherical harmonic expansions is similar to the previous oneforFourier series. Let ?2 = {x2 + x\ -fx\ = 1} be the two-dimensional sphere inR3, with euclidean surfacemeasure dx. The restriction to the sphere of the Laplaceoperator in space has eigenvalues {n(n + 1)}, 0 < n < +00, and an orthonormalcomplete system of eigenfunctions {^n,j(x)}j 0 < n < +00, 1 < j < 2n -j- 1,restriction to the sphere of harmonic homogeneous polynomials of degree n. If1 dnPn(z) = -7z^?y~r~^ (z2 ~ l) is the nth Legendre polynomial, then?1 ih.az

2n+l _ 2-4-1J2 ^n,j(x)^n,j(y) = ?^-Pn(x '?)J=l

Hence the spherical harmonics expansion of a distribution on the sphere is+00 2n+l r /.

n=0 i=l L^2 J=S ~^r~ / Pn(x 'y)v(y)dy

By Theorem 8, if (p(y) and A~1^2(p(y) are integrable functions, then the lastseries converges pointwise. In particular, the spherical harmonics expansion of theunit mass concentrated at x isHy-x) =^2^l Pn{y-z)6{z-x)dz =^2^?Pn{x.y).

Definen=0 "n=0

ipy) A-^S (y-x) = J2 A ???Pn(x y).Since -Pn(l)

= 1 and Pn(?1)=

(?l)n, the terms of the series at ?x do notconverge to zero. More precisely, at the point x the series diverges to infinityand this is natural, since (p(y) a singularity of type \y? x\~ . At the point ? x,where the function is smooth, the series oscillates. This phenomenon is due to thesingularities of the function expanded that propagate from one point focus at theantipodal point.In conclusion, the critical indexes forRiesz summability on euclidean spaces, ontori and spheres are all different.The final remark is that, although up to here we were mainly concerned aboutfunctionswith integrable derivatives, it is also possible and it is somehow more natural tomeasure smoothness using square norms. In particular, in [18], combining theRademacher-Menchoff criterion on almost everywhere convergence of orthogonal series together with theWeyl estimates on distribution of eigenvalues, it is shown thatthe eigenfunction expansions of functions in Sobolev classes (1+ A)~a/ L2(M., dx)with a > 0 converge almost everywhere. Following [2]and [22], it is also possible toestimate the capacity of the divergence sets. The following theorem and its proofare essentially a mix of ideas in [18] and [17].



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Theorem 15. Let us introduce theBessel potentials,(1+ A)-a/2?(x) =? (1+ (2ttA)2P/2 V>a(s).AMoreover, let us define thea capacity of a set X inAi,

Ca(X)=mf? ? \v(x)\2dx,with1 Aya/2n(x)> 1 in \ .Then, if > 0,

E(^V>a)a(z)a < x eM, sup AA< cA-2Y>g2 (2+ (2ttA)2)1+ (2ttA)2)q\(x)\2A

In particular, if log (2+ A) (1+ A)a/ 0 are positive. This implies thatsup?>0 v;((i+A)-a/2M,^)v>A(z)A


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Finally, the pointwise convergence of the eigenfunction expansion follows fromthe boundedness of the maximal partial sums by a standard argument. DObserve that the above result holds for every eigenfunction expansion and it isindependent from the underlying manifold.

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dlpartimento di matem?tica, universita di mllano-blcocca, edificio u5, via r.cozzi53, 20125 Milano, ItaliaE-mail address : leonardoQmatapp. unimib. it

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Fourier Expansions of Functions with Bounded Variation of Several Variables

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