On Cesáro summability of fourier series with respect to double complete orthonormal systems

Abstract

Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E ₀ ⊂ I ² is any Lebesgue measurable set such that μ₂ E ₀ > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L¹(I ²) and a set E ^′₀ , ⊂ E ₀, μ₂ E ^′₀ > 0, such that

$$\int_{I^2 } {\phi (|f(x,y)|)dxdy < \infty } $$

and the sequence of double Cesàro means of Fourier series of f with respect to the system {ϕn(x)ϕm(y): n,m = 1, 2,...} is unbounded in the sense of Pringsheim (by rectangles) on E ^′₀ . This statement gives critical integrability conditions for the Cesàro summability a.e. of Fourier series in the class of all complete orthonormal systems of the type {ϕ n(x)ϕm(y): n,m = 1, 2,...}.