On the order of growth of rectangular partial sums of double orthogonal series

We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems. South-Ukrainian Pedagogical University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1299–1310, October. 1999.     

On the order of partial sums of general orthogonal series

It is shown that for convergence of every orthonormal system _n(x) given on [0, l],it is necessary and sufficient that, under the condition on tlie increasing function W(x) and for there hold almost everywhere on [0, 1].Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 451–462, October, 1969.     

On local equivalence of the majorant of partial sums and Paley function of Franklin series

In this paper we prove that the majorant of partial sums and the Paley function of Franklin series have equivalent norms in the space L _p (I), p > 0, provided that the “peak” intervals of Franklin functions with non-vanishing coefficients lie in I. Examples of series emphasizing that this condition is essential are also given.     

Trigonometric polynomial approximation in theL 1-norm

Ohne Zusammenfassung     

Strong approximation by rectangular partial sums of double orthogonal series

Пусть {? _ik(x):i, k=1, 2,...} — орто нормированная систе ма в пространстве с полож ительной мерой и {a _ik} — последов ательность действит ельных чисел, для которой $$\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik}^2 \kappa ^2 (i,k)< \infty ,$$ где {x(i, K)} — определенна я неубывающая последовательность положительных чисел. Тогда суммаf(x) двойног о ортогонального ряд а \(\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik} \varphi _{ik} (x)\) существует в смысле с ходимости в метрикеL ² и сходимос ти почти всюду. Изучае тся порядок так называем ой сильной аппроксимац ииf(x) (при коэффициентн ых условиях) прямоуголь ными частными суммами \(s_{mn} (x) = \mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik} \varphi _{ik} (x)\) . Основной ре зультат состоит в сле дующем. Если {λ_j(m):m=1, 2,...} — неубывающи е последовательност и положительньк чисел, стремящиеся к ∞ и такие, что \(\mathop {\lim \sup }\limits_{m \to \infty } \lambda _j (2m)/\lambda _j (m)< \sqrt 2 \) дляj=1,2, и если $$\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik}^2 \left[ {\log log (i + 3)} \right]^2 \left[ {\log log (k + 3)} \right]^2 (\lambda _1^2 (i) + \lambda _2^2 (k))< \infty ,$$ TO ПОЧТИ ВСЮДУ $$\left\{ {\frac{1}{{mn}}\mathop \sum \limits_{i = 1}^m \mathop \sum \limits_{\kappa = 1}^m \left[ {s_{ik} (x) - f(x)} \right]^2 } \right\}^{1/2} = o_x (\lambda _1^{ - 1} (m) + \lambda _2^{ - 1} (n))$$ при min (m, n) → ∞.     

Rate of approximation by rectangular partial sums of double orthogonal series

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Dedicated to Professor K. Tandori on his seventieth birthday

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This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.     

Approximation of functions by partial sums of Fourier series in polynomials orthogonal on an interval

For certain weight functions p(t) and q(t), upper bounds are obtained for the difference between partial sums of Fourier series of a function/ with respect to the systems _p and _q of polynomials orthogonal on [–1, 1] (a comparison theorem is incidentally proved for the systems _p and _q). By using these upper bounds, known asymptotic expressions for the Lebesgue function, and an upper bound (forf W^r H) of the remainder in a Fourier-Chebyshev series, we establish corresponding results for Fourier series with respect to a system _p.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 431–441, October, 1970.     

On the growth order of partial sums of non-orthogonal series

a _k f _k , f _k L ₂, w-, (2), w(n) — . a _k f _k N {a _k }l ₂, {a _k }l ₂ ( 1, 2, 1a, 2a). ( 2) [8]. , {a _k } w-.     

On the partial sums of Vilenkin-Fourier series

The aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. Also, the obtained results we use to prove approximation and strong convergence theorems on the martingale Hardy spaces H _p , when 0 < p ≤ 1.     

On the growth order of the rectangular partial sums of multiple non-orthogonal series

Пустьd-натуральное ч исло,Z ^d — множество на боров k=(k ₁, ...,k _d ), состоящих из неотрицательных цел ыхk _j ,Z ₊ ^d =k∈Z ^d :k≧1. Предположи м, что системаf _k (x):k∈Z ₊ ^d ? ?L₂(X,A, μ) и последовател ьностьa _k :k∈Z ₊ ^d . таковы, чт о для всех b∈Z^d и m∈Z ₊ ^d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z ₊ ^d положительн а и не убывает. Например, есл иf _k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {a_k}. В работе получены оце нки порядка роста пря моугольных частных суммS _m (x)= =∑ a_kf_k(x) при maxmj→∞ как в случ ае {a_k}∈l₂, таки для {a_k}l₂. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными.     

Trigonometric series with positive partial sums

Translated from Matematicheskie Zametki, Vol. 53, No. 3, pp. 149–152, March, 1993.     

The order of growth of quadratic partial sums of a double Fourier series

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 69–79, June, 1992.     

Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums

It is proved that a trigonometric cosine series of the form _n ^Emphasis>=0/ a _n cos(nx) with nonnegative coefficients can be constructed in such a way that all of its partial sums are positive on the real axis. It converges to zero almost everywhere and is not a Fourier-Lebesgue series. Some other properties of trigonometric series with nonnegative partial sums are also studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 24–41, January, 1996.