G. NADIBAIDZE. a n. n=1. n = 1 k=1

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1 GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No., 999, 83-9 ON THE ABSOLUTE SUMMABILITY OF SERIES WITH RESPECT TO BLOCK-ORTHONORMAL SYSTEMS G. NADIBAIDZE Abstract. Theorems determining Weyl s multipliers for the summability almost everywhere by the c, method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence {ωn)} is the Weyl multiplier for the summability almost everywhere by the c, method of all orthogonal series, then there exists a sequence {N k } such that {ωn)} will be the Weyl multiplier for the summability almost everywhere by the c, method of all series with respect to the k -orthonormal systems. The present paper deals with the summability almost everywhere a.e.) by the c, α method of series with respect to block-orthonormal systems. Under the summability by the c, α method of the series a n is understood the convergence of the series where α) σ σ α) n = A α n n σα) n, A α n k a k k= are the Cesàro c, α)-means. The problem of the summability a.e. by the c, α method of orthogonal series was considered by P.L. Ul yanov []. In particular, he proved that if 99 Mathematics Subject Classification. 4C. Key words and phrases. Block-orthonormal systems, Weyl multiplier, series with respect to block-orthonormal systems, summability almost everywhere by the c, method X/99/-83$5./ c 999 Plenum Publishing Corporation

2 84 G. NADIBAIDZE the condition <, ) n ωn) is fulfilled for a positive nondecreasing sequence {ωn)}, then the convergence of the series an ωn) guarantees the summability a.e. on, ) by the c, α method α > ) of the series a n ϕ n x) ) for every orthonormal system from L, ). If however n ωn) =, then there exists an even function fx) L p [, π] such that its Fourier p series fx) c n cos nx converges a.e. on [, π] and for every fixed α > is not c, α summable a.e. on [, π] though c n ωn) <. Definition see []). Let {N k } be an increasing sequence of natural numbers, k = N k, N k ], k =,,..., and {ϕ n } be a system of functions from L, ). The system {ϕ n } will be called a k -orthonormal system k -ONS) if: ) ϕ n =, n =,,... ; ) ϕ i, ϕ j ) = for i, j k, i j, k. Definition see []). A positive nondecreasing sequence {ωn)} will be called the Weyl multiplier for the summability a.e. of series with respect to the k -ONS {ϕ n } if the condition a n ωn) < 3)

3 ON THE ABSOLUTE SUMMABILITY OF SERIES 85 guarantees the summability a.e. by the c, α method of the corresponding series ). Below we shall quote the theorem showing that if the sequence {ωn)} is the Weyl multiplier for the summability a.e. by the c, method of all orthogonal series ), then it will be the Weyl multiplier for the summability a.e. by the c, method of all series ) with respect to the k -ONS for the increasing sequence of natural numbers {N k }. Theorem. If a positive nondecreasing sequence {ωn)} is the Weyl multiplier for the summability a.e. by the c, method of all orthonormal series ), then there exists an increasing sequence of natural numbers {N k } such that {ωn)} is the Weyl multiplier for the summability a.e. by the c, method of all series ) with respect to the k = N k, N k ]-ONS. Proof. We prove this theorem by the Wang Ul yanov s scheme see []) modifying it accordingly. Let the positive nondecreasing sequence {ωn)} be the Weyl multiplier for the summability a.e. by the c, method of all orthogonal series ). Then condition ) is fulfilled. As is known see []), for the positive nondecreasing on [n, ) function ωx) the series m ωm) m=n and m=n m ω m) converge or diverge simultaneously. Therefore, taking into account ), we have n ω 4 n) <. Then where Rn) = Rn) n ω 4 <, 4) n) k= k=n k ω 4 k) k ω 4 k). Obviously, R) =, Rn) < Rn ) and lim n Rn) =. Define the sequence kn) by the recursion formula { kn) if Rn ) kn), k) =, kn ) = n. kn) if Rn ) < kn),

4 86 G. NADIBAIDZE Thus we obtain the nondecreasing sequence of nonnegative integers for which kn) Rn), n =,,.... 5) Note that for the sequence kn) there exists an increasing sequence of natural numbers {N k } it is assumed that N = ) which is defined by the formula kn) = max{k : N k < n}. Then, taking into account 4) and 5), we find that kn) n ω 4 <. 6) n) Let {ϕ n } be a block-orthonormal system with k = N k, N k ] and condition 3) be fulfilled. Then for the corresponding series ) we have σ n x) σ n x) = nn ) a i i ) ϕ i x), n. Denoting by c the absolute positive constants which, generally speaking, may have different values in different inequalities and using 6), we find that c n= n c n= i= σn x) σ n x) dx c N kn) c i= n= σ n x) σ n x) dx n n a i i )ϕ i x) dx i= a i i )ϕ i x) N kn) n kn) a i i n kn) n i=[ 4 n] n i ai i= i a i i= c n i=n kn) n i=n kn) kn) ] c n a i i )ϕ i x) dx a i i [ [ 4 n] i= i a i kn) n 4 n n

5 ON THE ABSOLUTE SUMMABILITY OF SERIES 87 n 9 8 kn) nω 4 n)) n n nω 4 n) c c n 3 ω 4 n) n 3 i= n i=[ 4 n] i=[ 4 n] i a i i ai ωi) c c i a i i= ) whence by Levy s theorem σn x) σ n x) < a.e. on, ). n= kn) kω 4 n) ) a i ωi) <, The theorem below makes it possible to determine the Weyl multipliers for the summability a.e. of the series ) with respect to the k -ONS for regularly increasing sequences {N k }. Theorem. Let an increasing sequence of natural numbers {N k } be given, for which the condition is fulfilled, and let N k=n k = O n N n ) kn) = max{k : N k < n}. n ) 7) If for the positive nondecreasing sequence {ωn)} condition ) is fulfilled, then for every k -ONS {ϕ n } the condition a n ωn) kn) < 8) guarantees the summability a.e. by the c, method of the corresponding series ). Proof. Let conditions ), 7) and 8) be fulfilled. Then for the corresponding series ) we have n= c n σn x) σ n x) dx N kn) i= n= a i i )ϕ i x) dx σn x) σ n x) dx n i=n kn) a i i )ϕ i x) dx

6 88 G. NADIBAIDZE c n kn) n i a i i= i=[ 4 n] i a i c kn) ] c kn) nω 4 n)) n n nω 4 n) c c = c c i= kn) n 3 i= c c i a i ωi) i= N ki) i a i ωi) n=i n i=[ 4 n] [ [ 4 n] i=n kn)n 3 8 n i a i i a i ) i a i ωi) nω 4 n) i= ki) n 3 kn) n 3 = j=ki) ki) c c i ai ωi) i ki) ) i= N j=ki) j whence by Levy s theorem i= j N j n=n j n=n ki) ) n 3 n 3 ) c c i ai ωi)ki) <, σn x) σ n x) < a.e. on, ). n= Remark. In Theorem, the Weyl multipliers defined by conditions ) and 8) can be assumed to be exact on the set of sequences {N k } with condition 7) in the sense that if condition ) is violated, then one can construct a sequence {N k } for which condition 7) is fulfilled and also there exists a trigonometric series b n cos nx, which is nonsummable by the c, α method for almost all x [, π] for every fixed α > ) though b n ωn) kn) <.

7 ON THE ABSOLUTE SUMMABILITY OF SERIES 89 Indeed, let the condition n ωn) = be fulfilled for the sequence {ωn)}. We construct an increasing sequence of natural numbers {N k } in such a way that the condition N k ) β, k = O < β n ωn), be fulfilled and the sequence N k k be increasing. Clearly, condition 7) is fulfilled see [3], Remark ). Take k s k =, k =,,..., n ωn) and c m = m ωm) sm ) β Then for arbitrary ε m = ± we have = ε m c m ) ωm)km) = m= N k k= m=n k c N k k= m=n k On the other hand, n= { n m= n m= km) m ωm)s m ) β c m =,,.... km) m ωm)s m ) β = N k k= m=n k m ωm)s m β c c m } n= { n m= n m= =. ω n )s n β s Nk ) β m ωm)s m ) β <. m ωm)s m ) β } mωm)) s m β Therefore by Billard s theorem [], for almost all choices of ε k = ± the series ε m c m cos mx m=

8 9 G. NADIBAIDZE is c, α -nonsummable α > ) at almost every point x [, π] though b n ωn) kn) <, where b n = ε n c n. Remark. The above theorems remain also valid for c, α methods with cα >. References. P. L. Ul yanov, Solved and unsolved problems of the theory of trigonometric and orthogonal series. Russian) Uspekhi Mat. Nauk 9964), No., V. F. Gaposhkin, Series of block-orthogonal and block-independent systems. Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 99, No. 5, 8; English translation: Soviet Math. Iz. VUZ), 3499), No. 5, G. G. Nadibaidze, On some problems connected with series with respect to k -ONS. Bull. Acad. Sci. Georgia 4499), No., Received ) Author s address: J. Javakhishvili Tbilisi State University Faculty of Mechanics and Mathematics, University St., Tbilisi 3843 Georgia