Partial Sums Of Fourier Series Research Articles

An explicit evaluation of total variations of fourier seri approximations to functions of finite variation

AbstractEarlier, the authors attempted an explicit evaluation of the total variation of best piecewise polynomial approximations and obtained a result [9]. However, there, the discussion was on polynomials and cannot be applied to other families of functions. Thus, the present paper considers a similar discussion with respect to the trigonometric functions, that is, the Fourier series expansion. Specifically, the total variation of Fourier series approximations is investigated, and, in addition, the norms of Fourier coefficients are evaluated. For the norms of Fourier coefficients, we confirm that they converge to zero when the degree m → ∞ and its rate is O(m−1) (Riemann‐Lebesgue lemma). Conventionally, the norm of Fourier coefficients had been evaluated mostly by assuming the periodicity of functions to be approximated. However, in this paper, we evaluate the norm without using the periodicity condition of the approximated functions. Moreover, it is shown that the upper bound of the ratio between the total variations of K‐th partial sum of Fourier series and that of the function to be approximated, is proportional to K. This implies that compared to the evaluation of total variation of the best piecewise polynomial approximations obtained in the authors' previous paper, this evaluation excels it by far.

Summation of subsequences of partial sums of Fourier series

Summation of subsequences of partial sums of Fourier series

CONVERGENCE OF FOURIER SERIES ALMOST EVERYWHERE AND IN THE $ L$-METRIC

The following theorems are proved.Theorem 1. There exists a constant such that for any function there is a measurable function for which , and where is a partial sum of the Fourier series of , is a partial sum of the conjugate Fourier series, and is the conjugate function to .Theorem 2. For any function and there exists a measurable function such that , ( is Lebesgue measure), and both the Fourier series of and its conjugate series converge almost everywhere and in the metric of .Bibliography: 11 titles.

On an infinite linear combination of partial sums of Fourier series

On an infinite linear combination of partial sums of Fourier series

Summability Methods Fail for the 2 n th Partial Sums of Fourier Series

Summability Methods Fail for the 2 n th Partial Sums of Fourier Series

Summability methods fail for the 2ⁿ𝑡ℎ partial sums of Fourier series

Although the Fourier series of a continuous function need not converge everywhere, it was an important discovery of Fejér that this series must be Cesàro summable. Indeed, it is a frequent occurrence that convergence may be restored to an expansion by use of an appropriate summability method. What we show in this note is that the very opposite phenomenon can occur. Namely, that if one considers only the 2 n {2^n} th partial sums of the Fourier series, there is no summability method whatever which produces convergence for all continuous functions.

On restricted weak type (1,1)

Let { S k } k ≧ 1 {\{ {S_k}\} _{k \geqq 1}} be a sequence of linear operators defined on L 1 ( R n ) {L^1}({R^n}) such that for every f ∈ L 1 ( R n ) , S k f = f ∗ g k f \in {L^1}({R^n}),{S_k}f = f \ast {g_k} for some g k ∈ L 1 ( R n ) , k = 1 , 2 , ⋯ {g_k} \in {L^1}({R^n}),k = 1,2, \cdots , and T f ( x ) = sup k ≧ 1 | S k f ( x ) | Tf(x) = {\sup _{k \geqq 1}}|{S_k}f(x)| . Then the inequality m { x ∈ R n ; T f ( x ) > y } ≦ C y − 1 ∫ R n | f ( t ) | d t m\{ x \in {R^n};Tf(x) > y\} \leqq C{y^{ - 1}}\smallint _{{R^n}} {|f(t)|dt} holds for characteristic functions f (T is of restricted weak type (1, 1)) if and only if it holds for all functions f ∈ L 1 ( R n ) f \in {L^1}({R^n}) (T is of weak type (1, 1)). In particular, if S k f {S_k}f is the kth partial sum of Fourier series of f, this theorem implies that the maximal operator T related to S k {S_k} is not of restricted weak type (1, 1).

On certain linear combinations of partial sums of Fourier series

On certain linear combinations of partial sums of Fourier series

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 σp and σq of polynomials orthogonal on [−1, 1] (a comparison theorem is incidentally proved for the systems σp and σq). By using these upper bounds, known asymptotic expressions for the Lebesgue function, and an upper bound (forf e Wr Hω) of the remainder in a Fourier-Chebyshev series, we establish corresponding results for Fourier series with respect to a system σp.

The uniform convergence of fourier series in orthogonal polynomials

Bounds are obtained for the difference of partial sums of Fourier series for the functionf in two orthogonal systems, permitting a number of known results for Fourier-Chebyshev series to be carried over to Fourier series in certain systems of orthogonal polynomials.

On a theorem of A. N. Kolmogorov concerning lacunary partial sums of Fourier series

On a theorem of A. N. Kolmogorov concerning lacunary partial sums of Fourier series

On convergence and growth of partial sums of Fourier series

On convergence and growth of partial sums of Fourier series

Oscillation of partial sums of Fourier series

Oscillation of partial sums of Fourier series

Partial Sums of Fourier Series

Partial Sums of Fourier Series

Notes on Fourier analysis (XXVI): Lipschitz condition of partial sums of Fourier series

Notes on Fourier analysis (XXVI): Lipschitz condition of partial sums of Fourier series

The approximation by partial sums of Fourier series

uniformly in x as h->0, we shall say briefly thatf belongs to Lip a. Following a notation already used (see Zygmund [4](1)) we shall say that f belongs to lip oa, 0 0. It is a classical result of Lebesgue (see Zygmund [5, p. 61 1; hereafter this book will be denoted by T.S.) that if f belongs to Lip a and if sn denotes the nth partial sum of the Fourier series of f, then, uniformly

The Approximation by Partial Sums of Fourier Series

The Approximation by Partial Sums of Fourier Series

Note on the theory of series (XXIII): On the partial sums of Fourier series

1. In what follows f(θ) is a periodic function of L2,is the Fourier series of f(θ), andis the nth partial sum of T(θ). We denote by n(θ) any function of θ which is measurable, is finite p.p. †, and assumes non-negative integral values only, and by n(θ, H) an n(θ) none of whose values exceeds H. We shall sometimes write N for n(θ), and N(H) for n(θ, H), to simplify the set-up of formulae.

On the Partial Sums of Fourier Series at Points of Discontinuity

On the Partial Sums of Fourier Series at Points of Discontinuity

On the partial sums of Fourier series at points of discontinuity

On the partial sums of Fourier series at points of discontinuity