It is proved that if positive definite matrix functions (i.e. matrix spectral densities) S n , n=1,2,… , are convergent in the L 1-norm, ||Sn-S||L1® 0|S_{n}-S|_{L_{1}}to 0, and ò02plogdetSn(eiq) dq®ò02plogdetS(eiq) dqint_{0}^{2pi}log mathop{mathrm{det}}S_{n}(e^{itheta}),dthetatoint_{0}^{2pi}log mathop{mathrm{det}}S(e^{itheta}),dtheta, then the corresponding (canonical) spectral factors are convergent in L 2, ||S+n-S+||L2® 0|S^{+}_{n}-S^{+}|_{L_{2}}to 0. The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place.
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机译：证明如果正定矩阵函数（即矩阵谱密度）S n ，n = 1,2，…在L 1 范数上收敛，| | | S n -S || L 1 ®0 | S_ {n} -S | _ {L_ {1}}至0，和« 0 2p logdetS n （e iq ）dq®ò 0 < sup> 2p logdetS（e iq ）dqint_ {0} ^ {2pi} log mathop {mathrm {det}} S_ {n}（e ^ {itheta}），dthetatoint_ {0 } ^ {2pi} log mathop {mathrm {det}} S（e ^ {itheta}），dtheta，则相应的（规范）谱因子在L 2 中收敛，|| S + n -S + || L 2 ®0 | S ^ {+} _ {n} -S ^ {+} | _ {L_ {2}}至0。很容易看出，制定的对数条件对于后者的收敛是必要的。
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