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For given p ∈ [ 1 , ∞ ] and g ∈ L p ( R ) , we establish the existence and uniqueness of solutions f ∈ L p ( R ) , to the equation f ( x ) − a f ( b x ) = g ( x ) , where a ∈ R , b ∈ R ∖ { 0 } , and | a | ≠ | b | 1 / p . Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established. [ABSTRACT FROM AUTHOR]

This paper deals with the de la Vallée Poussin means of the triangular partial sums of multivariate Fourier series. We determine the exact order of the corresponding operator norms. The lower estimation of these norms will be extended to a class of projection operators having similar projection properties as the de la Vallée Poussin mean. [ABSTRACT FROM AUTHOR]

Abstract: We study the behavior of the Fourier sums in orthonormal polynomial systems, related to exponential weights on , in weighted and uniform metrics. [Copyright &y& Elsevier]

Abstract: In order to approximate functions defined on and having exponential singularities at the endpoints of the interval, we study the behavior of some modified Fourier Sums in an orthonormal system related to exponential weights. We give necessary and sufficient conditions for the boundedness of the related operators in suitable weighted -spaces, with . Then, in these spaces, these processes converge with the order of the best polynomial approximation. [Copyright &y& Elsevier]

Abstract: Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents a method for establishing direct and strong converse inequalities in terms of -functionals for convolution operators acting in homogeneous Banach spaces of multivariate functions. The method is based on the behaviour of the Fourier transform of the kernel of the convolution operator. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to prove some a.e. convergence results of Fejér and logarithmic means of subsequences of partial sums of Walsh–Fourier series of integrable functions. We prove for lacunary sequences that the means of the partial sums converges to a.e. Besides, for every convex tending to and every integrable function the logarithmic means of the partial sums converges to a.e. [Copyright &y& Elsevier]