Convergence of the Logarithmic Means of Two-Dimensional Trigonometric Fourier Series

Transcription

1 Bulletin of TICMI Vol. 0, No., 06, Convergene of the Logarithmi Means of Two-Dimensional Trigonometri Fourier Series Davit Ishhnelidze Batumi Shota Rustaveli State University Reeived April 8, 06; Revised June 0, 06; Aepted June 30, 06 Abstrat. We disuss on some onvergene and divergene properties of two-dimensional Nörlund logarithmi means of Fourier series. Keywords: Double Fourier series, logarithmi means, onvergene in norm. AMS Subjet Classifiation: 4A4.. Main Results Let f T, T = [, ] be a -periodi funtions with respet to eah variable. The two-dimensional Fourier series of f with respet to the trigonometri system is the series where s [f] = + m,n= fm, n e imx e iny, fm, n = 4 fx, ye imx e iny dxdy are the Fourier oeffiients of the funtion f. Let CT be the spae of ontinuous funtions are -periodi with respet to eah variable with the norm Let f CT. The expression f = sup fx, y. x,y T ωδ, f = sup { f + u, + v f, : u + v δ } Corresponding author. Davitishhnelidze@gmail.om

2 is alled the total modulus of ontinuity of the funtion f. The partial modulus of ontinuity are defined by Vol. 0, No., ω δ, f = sup { f + u, f, : u δ}, ω δ, f = sup { f, + v f, : v δ}. We also use the notion of a mixed modulus of ontinuity. They are defined as follows: { ω, δ, δ, f = sup f + u, + v f + u, f, + v + f, : u δ, v δ }, f CT. The Riesz s means of the Fourier series has been studied by a lot of authors. We mention for instane the papers of Szasz [] and Yabuta [], devoted to the logarithmi means. Similar means with respet to the Walsh and Vilenin systems were disussed by Simon [0], and Gat [5]. The Norlund logarithmi means has been studied in [-7],[0-]. In this paper we investigate the approximation properties of two-dimensional logarithmi means of double trigonometri Fourier series of f defined as follows: t n,m f, x, y = l n l m n i= m j= s i,j f, x, y n im j, l n = where S M,N f, x, y is the partial sum of double Fourier series of f defined by n =, M N s M,N f, x, y = m= M n= N fm, ne imx e iny. It is evident that where t n,m f, x, y fx, y = [fx + t, y + s fx, y] F n tf m sdtds, F n t = n D t l n n = and D tis Dirihlet ernel. For one dimensional trigonometri Fourier series Goginava and Tebuhava [6] proved that the following are true

3 50 Bulletin of TICMI Theorem A [6]. Let f CT and then ωδ, f = o log/δ t n f f oasn. Theorem B [6]. There exists a funtion f CT suh that ωδ, f log/δ and t n f, 0 diverges. It is well-nown that the following statement is true [3]. Theorem C Zhizhiashvili. Let f CT, then { S n,m f f ω n, f logn + + ω m, f logm + +ω, n, } m, f logn + logm +. From and Let A=a mnj denote a positive retangular matrix, i. e., a mnj =0 for j > m or > n, a a mnj > 0 for eah 0 j m, 0 n and m j=0 =0 For any double sequene S j, define t mn = m j=0 =0 n a mnj =. n a mnj s j, m, n = 0,,,... The sequene S j is said to be summable by A if t mn tends to a finite limit as m, n. A double retangular matrix A is said to be regular if it sums every bounded onvergent double sequene S j to the same limit. Neessary and suffiient onditions for the matrix A to be regular are nown see, e.g. [9]: lim m,n j=0 m a mnj = 0 = 0,,..., lim m,n =0 n a mnj = 0 j = 0,,...

4 Vol. 0, No., 06 5 Sine t n,m f f l n l m n i=0 m j=0 S i,j f f n im j, from and we onlude that the following theorem is true. Theorem. : Then Let f CT and ωδ, f = o. log/δ t n,m f f C 0 as m, n. In the paper we investigate sharpness of Theorem.. In partiular, the following is true Theorem. : and t n,n f, 0, 0 diverges. There exist a funtion f CT suh that ωδ, f, log/δ Proof : of Theorem. We hoose a monotonially inreasing sequene of positive integers{n ; } suh that n, n +, 3 l= n l n l < n n, 4 n n n i i=0 We onstrut a funtion f defined as follows. Set n i <. 5 fx, y = = f x f y n,

5 5 Bulletin of TICMI where f x = sin n + x [6 γn, 6 mn γ n ] x, =,,..., x [, ], where A is the harateristi funtion of a set A and mn n = max { s : sγ n γ n }, γn = 6 n + /.. First we prove that ωδ, f C. 6 log/δ For every suffiiently small δ > 0 there exists a positive integer suh that Sine from 3 and 4 we get n + / δ < n + /. f nl x + δ fx δ nl, l =,...,, fx + δ, y fx, y δ Consequently, l= log/δ Analogously, we obtain n l n + O l. n l= l ω δ, f C f nl x + δ f nl x + δ n + O n l= l. 7 log/δ

6 Vol. 0, No., ω δ, f C. 8 log/δ Sine ωδ, f C ω δ, f C + ω δ, f C from 7 and 8 we get 6 Next, we shall prove that t n, n f, 0, 0 diverges. It is lear that t n, n f, 0, 0 f0, 0 = t n, n f, 0, 0 = ft, sf n tf n sdtds f n t F n tdt n i= i f ni t F n tdt n i=+ i f ni tf n tdt = I II III. 9 Sine see [6] l nf nx = sinn + x sin x n = sin + x + + sin x/ + n n sinn + sin n x sin x/ sin x/ + sin n + x n 4 sin x/ n osn + x sin x/ = 3 sin n + x 4 sin x/ sin x,

7 54 Bulletin of TICMI we have I = n f n tf n tdt mn n +/ n +/ sin n + /t sint/ n i= ii + i + sin i + t sin t/ dt n n mn n +/ n +/ sin n + /t sin n t sint/ sin t/ dt n mn n +/ n +/ sin n + / t sin n + /t 4 sin t/ dt mn n +/ n +/ sin n + / t sin t/ dt mn n +/ n +/ sin n + / t os n + / t sin t/ n i= sin it i dt = = I I I 3 I 4 I 5. 0 It is evident that I, I 3, I 4, I 5 = 0 mn n +/ n +/ t dt = 0. n Sine see [4] sini + t i + t, i =,,..., n, for t I n, I n = n m= [α mn, β mn ],

8 Vol. 0, No., where α mn = m + 6 n + /, β mn = m n, m, n =,,... + / and for I we have sin n + / t /, n = sin x <, I n i + n i i + i + i= m= βm,n dt > 0. α m,n t Combining and we onlude that I > 0. 3 Now, we estimate II. Sine [6] t n f f ω /n, f log n + and ω f ni, n n i = 0, i =,,...,, n from 4 and 5 we get II C n i= i C i= It is obvious that t n f ni f ni C n i n i n n C n n i= n i i= ω f ni, n i n i n n 4 = o as. F n L log n log n n i= n i= D i n i log i + n i log n +.

9 56 Bulletin of TICMI Then we have III n i=+ i n n + F n n i=+ i 5 n n 4 n = o as. After substituting 3, 4 and 5 in 9 we obtain lim t n, n f, 0, 0 f0, 0 > 0. Referenes [] Gát G. and Goginava U., Uniform and L-onvergene of logarithmi means of Double Walh-Fourier series, Georgian Mathematial Journal,, 005, [] Gát G. and Goginava U., Uniform and L-onvergene of logarithmi means of Walsh-Fourier series, Ata Math. Sin. Engl. Ser.,, 006, [3] Gát G and Goginava U., Almost everywhere onvergene of a subsequene of the logarithmi means of quadratial partial sums of double Walsh-Fourier series. Publ. Math. Debreen, 7, - 007, [4] Gát G., Goginava U., Tebuhava G., Convergene in measure of logarithmi means of quadratial partial sums of double Walsh-Fourier series, J. Math. Anal. Appl. 33,, 006, [5] Gát G., Investigations of ertain operators with respet to the Vilenin system, Ata Math. Hungary, 6, - 993, 3-49 [6] Goginava U. and Tebuhava G., Convergene of the logarithmi means of Fourier series, Journal of Mathematial Analysis and Approximation Theory, 006, 30-4 [7] Goginava U. and Tebuhava G. Convergene of subsequenes of partial sums and logarithmi means of Walsh-Fourier series. Ata Si. Math. Szeged, 7, - 006, [8] Hardy G. H., Divergent Series, Oxford, at the Clarendon Press, 949 [9] Robison G. M., Divergent double sequenes and series, Trans. Amer. Math. So. 8, 96, [0] Simon P., Strong onvergene of ertain means with respet to the Walsh-Fourier series, Ata Math. Hungary, 49, , [] Száz O., On the logarithmi means of rearranged partial sums of a Fourier series, Bull. Amer. Math. So , [] Yabuta K., Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz s logarithmi means, Tohou Math. J., 970, 7-9 [3] Zhizhiashvili L., Trigonometri Fourier series and their onjugates, Kluwer Aad. Publishers, Dordreht, Boston, London, 996 [4] Zigmund A., Trigonometri series. and ed. Vol. J. Cambridge University Press, New yor, 959