We consider existence and uniqueness of symmetric approximation of frames bynormalized tight frames and of symmetric orthogonalization of bases byorthonormal bases in Hilbert spaces H . More precisely, we determine whether agiven frame or basis possesses a normalized tight frame or orthonormal basisthat is quadratically closest to it, if there exists such frames or bases atall. A crucial role is played by the Hilbert-Schmidt property of the operator(P-|F|), where F is the adjoint operator of the frame transform F*: H --> l_2of the initial frame or basis and (1-P) is the projection onto the kernel of F.The result is useful in wavelet theory.
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机译：我们考虑了Hilbert空间H中归一化紧框架的对称逼近和正交基的对称正交的存在与唯一性。更准确地说，如果存在这样的框架或基数，我们将确定给定框架或基数是否具有平方二次最接近的归一化紧框架或正交基。运算符（P- | F |）的Hilbert-Schmidt属性起着至关重要的作用，其中F是帧变换F *的伴随运算符：H->初始帧或基数的l_2和（1-P ）是F核的投影。此结果在小波理论中很有用。
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