Mathematical Notes, vol. 74, no. 1, 2003, pp. 132–135. Translated from Matematicheskie Zametki, vol. 74, no. 1, 2003, pp. 139–142. Original Russian Text Copyright c 2003 by V. G. Krotov. When is an Orthogonal Series a Fourier Series? V. G. Krotov Received July 1, 2002 Key words: Fourier series, orthonormal system, Orlicz space. Inhispaper[1](seealso[2,Chap.VI,Sec.4]),Orliczprovedconditionsnecessaryandsufficient for a series with respect to an orthonormal system (ONS) to be the Fourier series of a function from a certain class (see Remark 1 below). These conditions are expressed in terms of regular meanvaluesoforthogonalseries. Infact,theOrliczproofworksinamoregeneralsituation,which is described in this note. Suppose that X is a Banach space over the field of scalars F ( F = R or F = C), Φ = {φ k } ∞ k=1 ⊂ X is a sequence of elements from X , L Φ is the linear hull of this system, and X Φ is the closure of L Φ in the normed space X . We will assume that a countable set Φ ∗ = {φ ∗ k } ∞ k=1 and a function 〈· , ·〉 : X × Φ ∗ → F exist such that 〈· , ·〉 is linear and continuous in the first variable and satisfies the following condition 〈φ k ,φ ∗ i 〉 = δ ik , i, k ∈ N. (1) Condition (1) allows us to define the Fourier series x ∼ ∞  k=1 〈x, φ ∗ k 〉φ k , x ∈ X. Wesaythatcondition(A)isfulfilledif,foranyboundedsequence {p n }⊂L Φ ,thereexist x ∈ X and n j ↑∞ such that ∀i ∈ N lim j→∞ 〈p n j ,φ ∗ i 〉 = 〈x, φ ∗ i 〉. Condition (A) (which is an operational form of the condition “ X is a dual space”) is indeed, the technical kernel of the proof of the proposition given below, which provides a criterion for the property “the series ∞  k=1 a k φ k is a Fourier series” in terms of the matrix mean values of this series. Let T = {t nk }⊂ F be a matrix with finite strings (i.e., for each n ∈ N an m n ∈ N exists such that t nk =0for k>m n ). Later on, we use the following condition: ∀k ∈ N ∃ lim n→∞ t nk = t k , lim k→∞ |t k | > 0. (2) 132 0001-4346/2003/7412-0132$25.00 c 2003 Plenum Publishing Corporation