On cubic lacunary Fourier series

For , we analyze the behavior, near the rational points , of , considered as a function of . We expand this series into a constant term, a term on the order of , a term linear in , a ``chirp" term on the order of , and an error term on the order of . At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when , both the real and imaginary parts of the cubic series are differentiable almost nowhere.

     

A summability method for the arithmetic Fourier transform

The Arithmetic Fourier Transform (AFT) is an algorithm for the computation of Fourier coefficients, which is suitable for parallel processing and in which there are no multiplications by complex exponentials. This is accomplished by the use of the Möbius function and Möbius inversion. However, the algorithm does require the evaluation of the function at an array of irregularly spaced points. In the case that the function has been sampled at regularly spaced points, interpolation is used at the intermediate points of the array. Generally theAFT is most effective when used to calculate the Fourier cosine coefficients of an even function.In this paper a summability method is used to derive a modification of theAFT algorithm. The proof of the modification is quite independent of theAFT itself and involves a summation by primes. One advantage of the new algorithm is that with a suitable sampling scheme low order Fourier coefficients may be calculated without interpolation.     

Summability of Fourier series with the method of lacunary arithmetical means at the Lebesgue points

The existence of the `rare' sequence of partial sums summable with the method of arithmetical means at each Lebesgue point is proved in the paper. The proof is based on the strategy of random choice.

     

On the harmonic summability of Fourier series

Proceedings - Mathematical Sciences -     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.     

On the harmonic summability of double Fourier series

Summary The author studies a problem relating Harmonic Summability of doubleFourier serie analogous to theGergen's criteria for convergence.     

On the absolute summability of factored Fourier series

Summary In this paper the author has considered the harmonic summability of the factored Fourier series when ϕα(t), α > 0, is of bounded variation.     

On application of matrix summability to Fourier series

Bor has recently obtained a main theorem dealing with absolute weighted mean summability of Fourier series. In this paper, we generalized that theorem for summability method. Also, some new and known results are obtained dealing with some basic summability methods.