Two problems in prediction theory

A prediction problem in $L^2 (w)$

“…This is the so-called (n+1)-step prediction variance. For the index set S 1 = S 0 ∪ {1, 2, · · · , n}, which corresponds to adding the next n frequencies to S 0 , it is shown in Nakazi [10] that…”

“…[2], [7], [10], [14]) are equivalent to finding the distance from the constant function 1 to a subspace M(S) = sp{e k : k ∈ S} in L p (w), where S is a subset of the integers Z, e k = e −ikλ , w is a nonnegative integrable function on the unit circle T , 0 < p < ∞, and L p (w) is the weighted L p space on T with norm f p = { T |f | p wdµ} 1/p . Here µ is the Lebesgue measure on T , so normalized that µ(T ) = 1.…”

“…In this case, the well-known Szegö theorem asserts that, for p > 0, (1.1) σ p (w, S 0 ) = exp 1 p T log wdµ if log w ∈ L 1 ; otherwise σ p (w, S 0 ) = 0 (see, e.g., Gamelin [5, p. 156]). The work in Nakazi [10] for the index set S 1 = S 0 ∪ {1, 2, . .…”

“…For i = 0, the (n + 1)-step prediction problem has been solved [1], [10] under the additional assumption that P n (z) = n k=0 c k z k = 0, for all |z| < 1, where the c k 's are defined by…”

Some extremal problems in 𝐿^{𝑝}(𝑤)

“…This notion has been of considerable interest in the theory of stationary processes and harmonic analysis (see [4,6,7,9]). This paper is concerned with evaluating σ p (w, S), and exploring its relationship to other constructions.…”