Partial Fourier sum

$$ \newcommand{\vb}[1]{\mathbf{#1}} $$ A partial sum of the Fourier series of a given function.

In the classical one-dimensional case where a function $f$ is integrable on the segment $[-\pi,\pi]$ and

$$S[f] = \frac{a_0}{2} + \sum_{k=1}^{\infty} (a_k \cos(k x) + b_k \sin(kx))$$

is its trigonometric Fourier series, the partial Fourier sum $S_n(f ; x)$ of order $n$ of $f$ is the trigonometric polynomial

$$S_n(f ; x) = \frac{a_0}{2} + \sum_{k=1}^{n} (a_k \cos(k x) + b_k \sin(kx))$$

With the use of the sequence of partial sums $S_n(f; x)$, $n=1, 2, \ldots$, the notion of convergence of the series $S[f]$ is introduced and its sum at a point $x$ is defined as follows:

$$ S(x) = \lim_{n\to\infty} S_n(f; x). $$

At every point $x$, the Dirichlet formula

$$ S_n(f; x) = \frac1\pi \int_{-\pi}^\pi f(x+t) D_n() \, dt, \qquad n=0,1,\ldots, $$

is true; here,

$$ D_n(t) = \frac12 + \sum_{k=1}^n \cos kt = \frac{\sin(n+\frac12)t}{2\sin \frac t2} $$

is the Dirichlet kernel of order $n$. This formula plays a key role in many problems in the theory of Fourier series.

If a series $S[f]$ is given in complex form, i.e., if

$$ \begin{gathered} S[f] = \sum_{k\in\Z} c_k e^{ikx}, \\ c_k = \frac{1}{2\pi} \int_{-\pi}^\pi f(t) e^{-ikt} \, dt, \end{gathered} $$

where $\Z$ is the set of all integers, then

$$ S_n(f; x) = \sum_{|k| \le n} c_k e^{ikx}. $$

In the multi-dimensional case, a notion of partial sum can be introduced in numerous different ways, none of which can be regarded as preferable.

One of the possible general approaches is as follows: Let $\R^N$ be the $N$-dimensional Euclidean space of points (vectors) $\vb{x} = (x_1, \ldots, x_N)$, and let $\Z^N$ be the integer lattice in $\R^N$, i.e., the set of vectors $\vb{n} = (n_1, \ldots, n_N)$ with integer coordinates. For vectors $\vb{x}, \vb{y} \in \R^N$, let

$$ (\vb{x}, \vb{y}) = x_1y_1 + \cdots + x_N y_N, \qquad |\vb{x}| = \sqrt{(\vb{x}, \vb{x})}. $$

Further, let

$$ Q_N = \{ \vb{x} \in \R^N : -\pi \le x_k \le \pi, k = 1, \ldots, N\}, $$

let $f(\vb{x}) = f(x_1, \ldots, x_N)$ be a function that is $2\pi$-periodic in each variable $x_k$ and integrable over a cube $Q_N$, and let

$$ \begin{gathered} S[f] = \sum_{\vb{k}\in\Z^N} c_{\vb{k}} e^{i (\vb{k},\vb{x})}, \\ c_{\vb{k}} = \frac{1}{(2\pi)^N} \int_{Q_N} f(\vb{t}) e^{-i (\vb{k},\vb{t})} \, d\vb{t}, \end{gathered} $$

be its Fourier series.

Further, let $\{G_\alpha\}$ be a family of bounded domains in $\R^N$ that depend on a real parameter $\alpha$ and are such that any vector $\vb{n} \in \R^N$ belongs to all domains $G_\alpha$ for sufficiently large $\alpha$. In this case, the expression

$$ S_\alpha(\vb{x}) = S_{G_\alpha} (f; \vb{x}) = \sum_{\vb{k} \in G_\alpha \cap \Z^N} e^{i(\vb{k},\vb{x})} $$

is called a partial Fourier sum of the function $f$ corresponding to the domain $G_\alpha$, and the expression

$$ D_{G_\alpha}(\vb{t}) = \frac{1}{2^{N}} \sum_{\vb{k} \in G_\alpha \cap \Z^N} e^{-i(\vb{k}, \vb{t})} $$

is called the Dirichlet kernel corresponding to this domain. It is clear that, for any vector $\vb{x} \in \R^N$, the following formula holds:

$$ S_{G_{\alpha}}(f; \vb{x}) = \frac{1}{\pi^N} \int_{Q_N} f(\vb{x} + \vb{t}) D_{G_\alpha} (\vb{t}) \, d\vb{t}. $$

This definition allows one to consider the problem of the convergence (or summability) of the series $S[f]$ as $\alpha\to\infty$. By virtue of the boundedness of the domains $G_\alpha$ the expression for $S_\alpha(\vb{x})$ is always a trigonometric polynomial.

The cases where $N$-dimensional spheres or $N$-dimensional intervals centred at the origin are taken as $G_\alpha$ are most often encountered and are well studied. The expressions

$$ \sum_{|\vb{k}| \le \alpha} c_{\vb{k}} e^{i (\vb{k}, \vb{x})}, \qquad \alpha > 1, $$

are called spherical partial sums, and the expressions

$$ \sum_{|k_j| \le n_j} c_{\vb{k}} e^{i (\vb{k}, \vb{x})}, $$

where $\vb{n} = (n_1, \ldots, n_N)$ is an arbitrary vector from $\Z^N$ with positive coordinates, are called rectangular partial sums. In recent years, in connection with problems in the approximation of functions from Sobolev spaces, partial Fourier sums constructed by "hyperbolic crosses" , namely, expressions of the form

$$ \begin{gathered} \sum_{\vb{k} \in \Gamma_{r,\alpha}} c_{\vb{k}} e^{(\vb{k}, \vb{x})}, \\ \Gamma_{r,\alpha} = \left\{ \vb{k} \in \Z^N : \prod_{i=1}^N |k_i|^{r_i} < \alpha, r_i > 0\right\}, \end{gathered} $$

have been extensively used. For Fourier series in general orthonormal systems of functions, partial Fourier series are constructed analogously. (Cf. also Orthonormal system.)

Various properties of partial Fourier sums and their applications to the theory of approximation and other fields of science can be found in, e.g., [a1], [a3], [a4], [a5], [a7]. [a6], [a2],

References

[a1]	N. Bary, "Treatise on trigonometric series" , 1; 2 , Pergamon (1964) Zbl 0129.28002
[a2]	A. Zygmund, "Trigonometrical series" , 1; 2 , Cambridge Univ. Press (1959)
[a3]	R. Edwards, "Fourier series: A modern introduction" , 1; 2 , Springer (1979)
[a4]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1; 2 , Springer (1963/70)
[a5]	W. Rudin, "Fourier analysis on groups" , Interscience (1962)
[a6]	G. Szegő, "Orthogonal polynomials" , Amer. Math. Soc. (1959)
[a7]	A. Stepanets, "Classification and approximation of periodic functions" , Kluwer Acad. Publ. (1995)