THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS


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1 Scientiae Mathematicae Japonicae Online, Vol. 5,, THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS CHIKAKO HARADA AND EIICHI NAKAI Received May 4, ; revised August, Abstract. Pinsky, Stanton Trapa [] showed, for example, that the spherical partial sum of the Fourier series or the Fourier transform of the characteristic function of the ndimensional ball x diverges at x when n. We point out that, in three dimensions, the square partial sum of it converges at x. ϕ be a function of bounded variation with compact support. We show that, for the radial function fx ϕ x, x Ê, the square partial sum of the Fourier transform converges to ϕ+ at x.. Introduction In one dimension, the behavior of Fourier series at a point depends only on the behavior of the function in a neighborhood of that point. In particular, if the function is ero on an interval, then the Fourier series converges to ero on that interval. However, in two or more dimensions, this localiation property does not generally hold. In two or more dimensions, there are many ways to add up the terms of the Fourier series; spherical partial sum, rectangular partial sum, square partial sum, etc. Pinsky, Stanton Trapa [] Kuratsubo [] [] investigated the convergence of the spherical partial sums of radial functions. Pinsky, Stanton Trapa [] showed, for example, that the spherical partial sum of the Fourier series or the Fourier transform of the characteristic function of the ndimensional ball x diverges at x when n. In this paper, we study the convergence of the square partial sums of radial functions in three dimensions. ϕ be a function of bounded variation with compact support. We show that, for the radial function fx ϕ x, x R, the square partial sum of the Fourier transform converges to ϕ+ at x. In the case that f is the characteristic function of the dimensional ball x, this convergence is equivalent to the following: lim d d d, + < where + +. Our result is for the Fourier transform. It is unknown whether the square partial sums of the Fourier series converges, though the square partial sum of the Fourier series of the characteristic function of the dimensional ball seems to converge by a numerical data. Our method is not valid for the case x. Goffman Liu [5] showed that the square partial sums of the Fourier series of a function f has the localiation property for f W n T n. However, the characteristic function of the dimensional ball x is not in W T. Mathematics Subject Classification. Primary 4B5. Key words phrases. Fourier transform, square partial sum, radial function, bounded variation.
2 Chikako HARADA Eiichi NAKAI In the next section we state definitions main results. We give a proof in the third section. To prove the main result we use the measure of the intersection of the cube {ξ R : ξ k, k,, } the ball {ξ R : ξ r}. For convenience we give a calculation of this measure in the fourth section.. Definitions main results Z n denotes the ndimensional integer lattice, whose points are written m m,..., m n, where m k are any integers. T n denotes the ndimensional torus, whose points are written x,..., x n, where x k. R n denotes the ndimensional Euclidean space, whose points are written x x,..., x n, for x, y R n, its inner product xy denotes n k x ky k. The Fourier series of a function F on T n, its spherical partial sum S sph partial sum S sq are defined by ˆF m n F xe imx dx, m Z n, T n S sph x ˆF me imx, m n m k, x T n, m < k S sq x ˆF me imx, x T n. m k <:k,,,n its square The Fourier transform of a function f on R n, its spherical partial sum f sph square partial sum are defined by ˆfξ n fxe iξx dx, ξ R n, R n f sph x ˆfξe iξx dξ, ξ n ξ k, x R n, ξ < k x ˆfξe iξx dξ, x R n. ξ k <:k,,,n its Pinsky, Stanton Trapa [] investigated the spherical partial sum of the Fourier series of the radial functions. Theorem A []. If <a, n, then the spherical partial sum of the ndimentional Fourier series of the characteristic function of the ball x a diverges at the center x. Theorem B []. F x ϕ x for x T, where <a, ϕr is a smooth function on the interval [,a] ϕr for r>a.if ϕa then the spherical partial sum of the Fourier series converges for all x T. Conversely, if the spherical partial sum of the Fourier series converges at x then ϕa. Theorem C []. fx ϕ x for x R, where ϕr has a continuous derivative on the interval [,a] ϕr for r>a. the spherical partial sum of the Fourier transform converges at x if only if ϕa.
3 The square partial sums of the Fourier transform of radial functions in three dimensions Pinsky, Stanton Trapa [] also stated a example with graphs as follows: { for x a, F x x T, <a, for x >a, ˆF a 6, ˆF m m a cos a m m + sin a m m, m,,. The spherical partial sum of this series for a, 4 x,x,x x,, has been graphed using Mathematica is depicted in Fig.. The partial sum for 55 is depicted in Fig Fig. : S sph x,,, Fig. : S sph x,,, 55 On the other h, the square partial sums for for are depicted in Fig. 4, respectively. We point out that the square partial sum seems to converge at x Fig. : S sq x,,, Fig. 4: S sq x,,, Our main results are as follows: Theorem.. fx ϕ x for x R, where ϕ is a function of bounded variation with compact support. ϕ+ as +.
4 Chikako HARADA Eiichi NAKAI Since ξ k <:k,, ˆfξ dξ fy R e iξy dξ ξ k <:k,, fy sin y sin y sin y R y y y dy, dy we have the following: Corollary.. ϕ be a function of bounded variation., for all a>, ϕ y sin y sin y sin y dy ϕ+ as +. y <a y y y ϕr a in Corollary.. we have the following: Corollary.. < d d d as +.. Proof We prove the theorem in this section. Lemma. Kuratsubo []. Suppose a > ϕ is a function of bounded variation. { ϕ x for x a, fx for x >a. we have a ˆfξ ϕa χˆ a ξ ˆχ t ξ dϕt, where { for x t, χ t x x R n, for x >t, dϕ is the LebesgueStieltjes measure generated by ϕ. In the case n, by Lemma[], P, we have t cos tr sin tr ˆχ t ξ Ht, ξ, where Ht, r + r r. supp ϕ [,a]., by Lemma., we have ˆfξ A ξ, where Ar ϕaha, r a Ht, r dϕt. V r be the measure of the intersection of the cube {ξ R : ξ k, k,, } the ball {ξ R : ξ r}. V r for r ˆfξ dξ ArV r dr ξ k <:k,, ϕa Ha, rv a r dr Ht, r dϕt V r dr.
5 The square partial sums of the Fourier transform of radial functions in three dimensions It, Ht, rv r dr. Since V r is continuous V r 4r for small r>, Ht, rv r is continuous on [,a] [, ]. By Fubini s theorem we have a ϕaia, It, dϕt. If we show that. It, is bounded It, { for t>, for t, as +, then we have ϕa ϕa ϕ+ ϕ+ as +. It remains only to prove.. By elementary calculations see Section 4 for example, we have V r for <r<, V r V r for r<, V r V r+v r for r<, where V r 4 r, V r 8r +6r, V r 8r +6r, V r 8 + r 4r arctan + r +8 arctan +8r arctan + r + r + r 8r arctan + r r + r, + r V r 4r, V r r 8r, V r r 8r, r 48r arctan + r +4r arctan + r + r + r V 4r arctan + r r + r It, I t, +I t, +I t, +I t,,.
6 4 Chikako HARADA Eiichi NAKAI I t, I t, Ht, rv r dr, I t, Ht, rv r dr, I t, Ht, rv r dr, Ht, rv r dr. we have. I t, t cos tr r. I t, +I t, [ sin tr + r 6t cos tr r ] 6 sin tr + r 4r dr t cos tr + t cos tr r sin tr sin t dr + t d, r + sin tr r 8r r dr 6 sin tr 4t cos tr 4 sin tr + + dr r r 4t cos tr dr 4 t d t sin sin t t By the change of variables, u +r /, we have I t, where H t t, +u V +u u du +u t +u u du + E u sin t +u du +u E ut cos E u r V r 4 +u arctan u arctan +u + u +u arctan E u r V r u E u u +u +u. Since E / 4/ E, we have I t, 4 sin t where t d. I t, +I t,, say, +u u +u, E u sin t +u du, E u 48 + u 4u + u arctan u. +u + u + u 4
7 The square partial sums of the Fourier transform of radial functions in three dimensions 5.4 G t, I t, Eu sin t +u du E E + E. 4 sin t + G t,. By.,..4 we have t t I t, d 4 d + G t,. t We note that It, 6 d + sup Eu C<+, u { t for t>, d as +, for t, t d as +. t For all ε> there exists δ> such that δ Eu sin t +u du δ sup Eu < ε. u By the change of variables v +u, using the RiemannLebesgue theorem, we have Eu sin t +u du v E v +δ v sin tv dv < ε, δ for large. Hence Gt, as +. Therefore we heve.. The proof is complete. 4. The measure of the intersection of the cube the ball For a measurable set Ω, we denote its measure by Ω. Q {x, y, R : x, y, }, B r {x, y, R : x + y + r }, V r V, r Q B r, gt V,t Q B t. V r V, r V,r/ gr/, V r g r/. In this section we calculate gt to get V r. g t <t, g t <t, gt g t <t<, g 4 t t.
8 6 Chikako HARADA Eiichi NAKAI We show that 4. g t 4 t, g t 8t +6t, g t 8t +6t +8 t 4t 8 arctan t +8t arctan t +t t 8t arctan t t t, g 4 r 8. If we can show this, then we have that g t 4t, g t 8t +t, g t 8t +t 48t arctan t +4t arctan t +t t 4t arctan t t t, g 4 r. Actually, since arctan v / +v, we have 8 t 4t 8 arctan t 6t t t 48t arctan t, 8t arctan t +t t 8t arctan t t t 6t t t +4t arctan t +t t 4t arctan t t t. Fig. 5 Fig. 6 are the graphs of gt g t, respectively Fig. 5: gt.5.5 Fig. 6: g t are clear. In the following we show
9 The square partial sums of the Fourier transform of radial functions in three dimensions 7 For the case t> see Fig. 7, let K t {x, y, B t : x>}. For the case t> see Fig. 8 Fig. 9, let L t {x, y, B t : x>,>}. g t B t 6 K t g t B t 6 K t + L t. y y x x Fig. 7: Q B t,<t< Fig. 8: Q B t, <t< x + t y ly x O x t y Fig. : ly, t y t, <t< Fig. 9: L t, <t< K t t Hence we have 4.. kx dx kx {y, R :x, y, K t }. t t x dx t t + for t>.
10 8 Chikako HARADA Eiichi NAKAI see Fig. L t t t ly {x, R :x, y, L t }. ly dy t ly dy for <t<, t y ly t y x dx for <t< t y t. Ru u u u + arctan. u we have R u u. Hence t y t y / t y t y x dx t y u du / t y t y t y R R t y t y T y t y y t y arctan t y arctan. t y arctan t y arctan t y y t y + t t + y arctan t y arctan t +ty t y arctan t ty. t y T y t y arctan t y arctan, t y ly T y t y +. Now we have t T y dy T t T t t + arctan t + t arctan t +t t arctan t t t,
11 The square partial sums of the Fourier transform of radial functions in three dimensions 9 t Hence L t t y dy t / t t u du t t R R t + t arctan t. t t t ly dy T y dy t t y dy + t dy t t arctan t + arctan t + t arctan t +t t t arctan t t t. we have Acknowledgement The authors would like to thank Professor Shigehiko Kuratsubo for his many suggestions. References [] S. Kuratsubo, On pointwise convergence of Fourier series of indicator function of n dimentional ball, Sci. Report Hirosaki 4 996, [] S. Kuratsubo, On pointwise convergence of Fourier series of radial functions in several variables, Proc. Amer. Math. Soc , [] M. A. Pinsky, N. K. Stanton P. E. Trapa, Fourier series of radial functions in several variables, J. Funct. Anal. 6 99,. [4] M. A. Pinsky, Pointwise Fourier inversion related eigenfunction expansions, Comm. Pure. Appl. Math , [5] C. Goffman F. C. Liu, On the localiation property ouare partial sums for multiple Fourier series, Studia Math , Chikako Harada: Hyogo Prefectural Arima High School, 5 Tenjin, Sa City, Hyogo 6695, Japan address: Eiichi Nakai: Department of Mathematics, Osaka Kyoiku University, Kashiwara, Osaka , Japan address:
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