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SOME EXTREMAL PROBLEMS IN Lp(w)

SOME EXTREMAL PROBLEMS IN Lp(w),R. CHENG,A. G. MIAMEE,M. POURAHMADI

SOME EXTREMAL PROBLEMS IN Lp(w)   (Citations: 7)
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Fix a positive integer n and 1 <p< 1. We provide expressions for the weighted Lp distance These distances are related to other extremal problems, and are shown to be positive if and only if log w is integrable. In some cases they are expressed in terms of the series coecients of the outer functions associated with w.
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    • ...His result and technique have spawned considerable research in this area in the last two decades; see Miamee and Pourahmadi (1988), Miamee (1993), Cheng et al. (1998), Frank and Klotz (2002), Klotz and Riedel (2002) and Bondon (2002)...
    • ...where S c is the complement of S in Z \ {0} and f −1 ∈ L1; see Cheng et al. (1998) and Urbanik (2000)...
    • ...The first occurrence of (5) seems to be in the 1949 Russian version of Yaglom (1963) for the case of deleting finitely many points from S ∞. Proof of (5), in general, like those of the main results in Nakazi (1984), Miamee and Pourahmadi (1988), Cheng et al. (1998), and Urbanik (2000), is long, unintuitive and relies on duality techniques from functional and harmonic analysis and requires f −1 ∈ L1 which is not natural for the index set S ...
    • ...However, the present form of the lemma does not seem to be useful for predicting infinite-variance- or Lp-processes (Cambanis and Soltani, 1984; Cheng et al., 1998)...

    Yukio Kasaharaet al. Duals of random vectors and processes with applications to prediction ...

    • ...[2, 7, 10, 14]) are equivalent to finding the distance from the constant function 1 to a subspace M(S )= sp{ek : k ∈ S} in Lp(w), where S is a subset of the integers Z, ek = e−ikλ, w is a nonnegative integrable function on the unit circle T ,0 <p< ∞, and Lp(w) is the weighted Lp space on T with norm � f � p = { � T |f |pwdµ}1/p...
    • ...To name some related contributions, let us mention here Cheng et al. [2], Frank and Klotz [4], Klotz and Riedel [6], Kolmogorov [7], Miamee and Pourahmadi [9], Pourahmadi [13, 14], and Urbanik [15]...
    • ...At present, the best known general result is Theorem 2 of Cheng et al. [2] which states that, for such an S, σp(w, S) is positive if and only if log w ∈ L1(dµ)...
    • ...The rather curious “inverse” relationship between the distances in (2.2) and (2.3), and also the need for the unnatural condition w−1 ∈ L1(dµ) were explained by establishing a duality between L2(w) and L2(w−1) as Banach spaces (see [9, 2]) and noting that the complement Sc1 = Z0 \ S1 of S1 in Z0 is...
    • ...Though the latter unnatural restriction can be weakened [2] to log w ∈ L1(dµ), the quantity σq(wr ,S c) might not be well-defined...
    • ...Fortunately, for the index set S1, this difficulty was resolved in [2, Theorem 3] using another dual extremal problem in [3] related to the projection of Lp onto the Hardy space H p...
    • ...The following identity which is a generalization of [2, Theorem 6] is of independent interest and curious so far as its relation with σ2 2(w, S0 − m) and σ2 2(w, ˜ S1), where ˜...
    • ...Using this result and the duality relation (2.4), σp(w, S1) is found in [2]...
    • ...It seems quite likely that the one-dimensional orthogonalization technique used in [2, Theorem 5] can be extended to the Lp(w) setting, and then using the duality relation (2.4), one can also compute σp(w, S2)...

    Mohsen Pourahmadiet al. A prediction problem in $L^2 (w)$

    • ...However, when 1 < p < 2, p-stationary processes do not even have a well-defined notion of covariance or spectrum, so that neither the spectral-domain nor the time-domain techniques are as effective as they have been for 2-stationary processes [1, 2, 5, 6]. The innovation processftg of fXtg is defined by t D Xt PHt 1 Xt ,w herePHt 1 Xt stands for the metric projection of Xt onto Ht 1 D spfXt 1; Xt 2;:::g in the norm of L p.; ; P/...
    • ...36 R. Cheng, A. G. Miamee and M. Pourahmadi [2]...

    R. Chenget al. On the geometry of Lp (μ) with applications to infinite variance proce...

    • ...The papers [Mi, CMiP] contain further completions of these results...
    • ...In contrast to these careful investigations, in the proofs of [C, Thm. 4.1] and of [CMiP, Thm...
    • ...Our approach is similar to that one presented in the proof of [CMiP, Th. 3] for the case q = 1. However, since we wish to compute not only the prediction error as done there but also the orthogonal projection, we give a complete proof of our generalized result...
    • ...for some n ∈ N. The univariate versions of the results below can be found at [CMiP, Th. 5 and 6]. Let Aj be the j-th Taylor coefficient of the outer function � which has the properties (5.2), j ∈ N0...
    • ...We omit the proof since (5.14) can be obtained by a straightforward generalization of the proof given for [CMiP, Th. 5] to the multivariate case, and (5.15) follows by some simple matrix computations from (5.14)...

    Michael Franket al. A duality method in prediction theory of multivariate stationary seque...

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