Bhatia, Rajendra

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Title Some Problems in Fourier Analysis and Matrix Theory(RecentDevelopments in Linear Operator Theory and its Applications)Author(s)Bhatia, RajendraCitation数理解析研究所講究録 (2005), 1458: 38-45Issue Date2005-12URL http://hdl.handle.net/2433/47913RightType Departmental Bulletin PaperTextversionpublisherKyoto University38Some Problems in Fourier Analysis and MatrixTheoryRajendra BhatiaIndian Statistical Institute7, S. J. S. Sansanwal MargNew Delhi - 110016.Email : rbh@isid.ac.inWe discuss some problems studied in diverse contexts but with a commontheme; the use of Fourier analysis to evaluate norms of som $\mathrm{e}$ specialmatrices.Let $\ovalbox{\tt\small REJECT}_{n}$ be the space of $n\mathrm{x}$ $n$ matrices. For $A\in\ovalbox{\tt\small REJECT}_{n}$ let$||A||= \sup\{||Ax|| : x\in \mathbb{C}^{n})||x||=1\}$ ,be the usual operator norm of $A$ . Let $A\circ X$ be the entrywise productof two matrices A and $X$ and let$||A||_{S}= \sup\{||A\circ X|| : ||X||=1\}$ .This is the norm of the linear map on $\ovalbox{\tt\small REJECT}_{n}$ defined as $X\mapsto A\circ X$ . Since$A\circ X$ is a principal submatrix of $A\otimes X$ , we have $||A\circ X||\leq||A\otimes X||=$$||A||||X||$ , and hence$||A||_{S}\leq||A||$ .1458 2005 38-4539Let $\lambda_{1},$ . . . $7\lambda_{n}$ be distinct real numbers and let$\delta=\min_{i\neq j}|\lambda_{i}-\lambda_{j}|$ .Let $H$ be the skew-symmetric matrix with entries $h_{r\epsilon}$ defined as$h_{rs}=\{\begin{array}{l}\mathrm{l}/(\lambda_{r}-\lambda_{s})r\neq s\mathrm{O}r=s\end{array}$ (1}Motivated by problems arising in number theory, Montgomery and Vaughan[5] proved the following.Theorem 1. The norm of the matrix H is bound ed as$||H||\leq c_{1}/\delta$ , (2)where$c_{1}$$= \inf\{||\varphi||_{L_{1}}$ : $\varphi$ $\in L_{1}(\mathbb{R})$ , $\varphi$ $\geq 0$ , and $\hat{\varphi}(\xi)=\frac{1}{\xi}$ for $|\xi|\geq 1\}.(3)$Here $\hat{\varphi}$ stands for the Fourier transform of $\varphi$ . Further,$c_{1}=\pi$ . (4)A very special case of this theorem is “Hilbert’s inequality” Let$\lambda_{j}=j_{7}$ $j=1,2$ , $\ldots$ . Then the (infinite) matrix $H$ defined by (1) iscalled the Hilbert matrix. Hilbert showed that $H$ defines a bounded40operator on the space $\ell_{2}$ and $||H||<2\pi$ . This was improved uponby Schur who showed that $||H||=\pi$ . Different proofs of this fact werediscovered by others, one using Fourier series by Toeplitz. (Matricesstructured as $H$ are now called Toeplitz matrices.)In particular, this shows that the inequality (2) with $c_{1}=\pi$ is sharp(in the sense that if it is to hold for all $n_{7}$ then no constant smaller than$\pi$ would work).Now suppose we have two real $n$ -tuples $\lambda_{1}$ , . . . , $\lambda_{n}$ and $\mu_{1}$ , . . . ’ $\mu_{n}$where for all $\mathrm{i}$ and $j$ we have $\lambda_{i}\neq\mu_{j}$ . Let$\delta$ $= \min_{i_{2}j}|\lambda_{i}-\mu_{j}|$ .Let $\mathrm{A}I$ be the matrix with entries $m_{rs}$ defined as$m_{rs}= \frac{1}{\lambda_{r}-\mu_{s}}$ . (5)Motivated by problems arising in perturbation theory, Bhatia, Davis andMclntosh [1] proved the following.Theorem 2. The norm $||M||_{S}$ is bounded as$||M||_{S}\leq c_{2}/\delta$ , (6)wheree2 $= \inf\{||\varphi||_{L_{1}}$ :: $\varphi\in L_{1}(\mathbb{R}))\hat{\varphi}(\xi)=\frac{1}{\xi}$ ffoorr $|\xi|\geq 1\}$ . (7)41The constant $c_{2}$ tad been evaluated earlier by Sz-Nagy [6] and wehave$c_{2}= \frac{\pi}{2}$ . (8)Note that the infimum in (7) is over a class of functions larger than theone in (3).It has been shown by McEachin [4] that the inequality (6) is sharpwith $c_{2}=\pi/2$ , and the extremal value is attained when the points $\{\lambda_{i}\}$and $\{\mu_{j}\}$ are regularly spaced.The resemblance between the two problems is striking and it is anatural curiosity to ask whether good expressions for the norms $||\mathit{1}’I||$and $||H||_{S}$ may be found to supplement what is known.In [1] the authors considered also the case where $\{\lambda_{i}\}$ and $\{\mu_{j}\}$ are$n$ -tuples of complex numbers with the same restriction as before, $\mathrm{v}\mathrm{i}\mathrm{z}.$ ,$\delta=\min_{i,j}|\lambda_{i}-\mu_{j},|>0$ .They proved the following.Theorem 3. Let $M$ be the matrix (with complex entries) defined as in(5). Then$||M||_{\mathit{8}}\leq c_{3}/\delta$ , (9)42where$c_{3}= \inf\{||\phi||_{L_{1}}$ : $\varphi\in L_{1}(\mathbb{R}^{2}),\hat{\varphi}(\xi_{1}, \xi_{2})=\frac{1}{\xi_{1}+\iota\xi_{2}}$, for $\xi_{1}^{2}+\xi_{2}^{2}\underline{>}1\}$ .(10)An attempt to calculate the constant $c_{3}$ was made by Bhatia, Davisand Koosis [2]. These authors first obtained another characterisation of$c_{3}$ . Let $C$ be the class of all functions $g$ on $\mathbb{R}$ that satisfy the followingconditions(i) $g$ is even,(ii) $g(x)=0$ for $|x|\geq 1$ ,(iii) $\int_{-1}^{1}g(x)$ $=1_{7}$(iv) $\hat{g}\in L_{1}(\mathbb{R})$ .The following theorem was proved in [2]theorem 4.$c_{3}=$ $\inf$ $|\hat{g}|$ : g $\in C\}$ . (ii)Using this the following estimate was derived in [2]$c_{3} \underline{<}\frac{\pi}{2}$ $\frac{\sin t}{t}dt<$ 2.90901 (10)43The constant $c_{2}$ occurs in another context called Bohr’s inequality.This says that if a function $f$ and its derivative $f’$ satisfy the followingconditions(i) $f\in L_{1}(\mathbb{R})\mathrm{l}$ $f’\in L_{\infty}(\mathbb{R})$ ,(ii) $\hat{f}(\xi_{\wedge})=0$ for $|\xi|\leq$ J.Then$||f||_{\infty} \leq\frac{c_{2}}{\delta}||f’||_{\infty}$ , (13)and the inequality is sharp.Attempts have been made to extend this result to functions of severalvariables. Hormander and Bernhardsson [3] have shown that if $f$ is afunction on $\mathbb{R}^{2}$ satisfying conditions akin to (i) and (ii) above, then$||f||_{\infty} \leq\frac{c_{3}}{\delta}||\nabla f||_{\infty}$ . (14)With this motivation they tried to evaluate C3. Like the authors of [2],they too first prove (11), and then use it more effectively to show that2.903887282 $<c_{3}<$ 2.90388728275228. (15)It would surely be of interest to find the exact value of $c_{3}$ , especiallysince the formulas (4) and (8) are so attractive.Some other problems remain open. The estimate (6) has been shownto be sharp by McEachin [4]. The question about (9) does not seem to44have been addressed. The matrix (5) when $\{\lambda_{i}\}$ and $\{\mu_{\dot{i}}\}$ are pointson the unit circle was considered in [1]. An extremal problem involvingFourier series instead of Fourier transforms as in (7) and (10) arises inthis case. This too has not been solved.References[1] R. Bhatia, C. Davis and A. Mclntosh, Perturbation of spectral sub-spaces and solution of linear operator equations, Linear AlgebraAppl, 52/53 (1983) 45-67.[2] R. Bhatia, C. Davis and P. Koosis, An extremal problem in Fourieranalysis with applications to operator theory, J. Funct. Anal, 82(1989) 138-150.[3] L. Hormander and B. Bernhardsson, An extension of Bohr’s In-equality, in Boundary Value Problems for Partial Differential Equa-tions and Applications, J. L. Lions and C. Baiocchi, eds., Masson,Paris, 1993.[4] R. McEachin, A sharp estimate in an operator inequality, Proc.Amer. Math. Soc, 115 (1992) 161-165.[5] H. L. Montgomery and R. C. Vaughan, Hilbert’s Inequality, J. Lon-don Math. Soc, 8 (1974) 73-8245[6] B. Sz.-Nagy, \"Uber die Ungleichung von H. Bohr, Math. Nachr. 9(1953) 225-259

Some Problems in Fourier Analysis and Matrix Theory(Recent Developments in Linear Operator Theory and its Applications)

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Some Problems in Fourier Analysis and Matrix Theory(Recent Developments in Linear Operator Theory and its Applications)

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