# Approximation of functions of several real variables

The case when the function $f$ to be approximated depends on two or more variables:

$$f (t) = f ( t _ {1} \dots t _ {m} ) ,\ \ m \geq 2 .$$

(See Approximation of functions.) In comparison with the one-dimensional case, the study of problems on the approximation of functions of $m$( $m \geq 2$) variables is highly complicated because of essentially new circumstances that arise related to dimension. First of all this applies to the domain on which the approximation is carried out. Simply-connected compact sets (in the one-dimensional case intervals) in $\mathbf R ^ {m}$( even in the plane) can have a large variety of shapes; therefore it becomes necessary to classify the domains according to, for instance, the smoothness properties of their boundaries. Complications also arise in the description of difference-differential properties of functions of $m$ variables. Generally speaking, these properties can be different in different directions, and their characterization ought to take into account both the geometry of the domain on which the function is defined and the behaviour of the function while approaching the boundary, so that the study of boundary properties of the function is of great importance. If one attempts to simplify the solution of an approximation problem by passing to a domain with a simpler structure, the problem that arises is to extend a function $f$ from a domain $Q \subset \mathbf R ^ {m}$ into a canonical domain $Q _ {1}$( for instance a parallelepiped or the whole space $\mathbf R ^ {m}$) containing $Q$ while preserving certain smoothness properties (see Extension theorems). This kind of problems is closely connected with imbedding theorems, and also with questions arising in the solution of the boundary value problems of mathematical physics.

An increase in the number of independent variables complicates, of course, also the tools of approximation, because when the dimension is increased, with it, e.g., the degrees of the polynomials increase. An algebraic polynomial of degrees $n _ {1} \dots n _ {m}$ in the variables $t _ {1} \dots t _ {m}$, respectively, has the form

$$\tag{1 } P _ {n _ {1} \dots n _ {m} } ( t _ {1} \dots t _ {m} ) = \ \sum a _ {k _ {1} \dots k _ {m} } t _ {1} ^ {k _ {1} } \dots t _ {m} ^ {k _ {m} } ,$$

where $a _ {k _ {1} \dots k _ {m} }$ are real coefficients. The summation is carried out with respect to $k _ \nu$, $\nu = 1 \dots m$, from 0 to $n _ \nu$, so that, e.g., the subspace of polynomials of degree 3 in each of the $m$ variables has dimension $4 ^ {m}$. Sometimes the overall or total degree $n$ of the polynomial is fixed; then the summation in (1) is carried out over all indices satisfying the inequality $0 \leq k _ {1} + \dots + k _ {m} \leq n$. A real trigonometric polynomial of degrees $n _ {1} \dots n _ {m}$ in the variables $t _ {1} \dots t _ {m}$ can be written in the form

$$T _ {n _ {1} \dots n _ {m} } ( t _ {1} \dots t _ {m} ) = \ \sum a _ {k _ {1} \dots k _ \nu } \mathop{\rm exp} \left ( i \sum _ {\nu = 1 } ^ { {m } } k _ \nu t _ \nu \right ) ,$$

where complex coefficients $a _ {k _ {1} \dots k _ {n} }$ with indices of opposite sign are complex conjugate, and the summation is carried out with respect to $k _ \nu$, $\nu = 1 \dots m$, from $- n _ \nu$ to $n _ \nu$. Such a polynomial can be represented in the form of a linear combination of all possible products of the form $\phi _ {k _ {1} } ( t _ {1} ) \dots \phi _ {k _ {m} } ( t _ {m} )$, where $\phi _ {k _ \nu } ( t _ \nu )$ is either $\sin k _ \nu t _ \nu$( $0 \leq k _ \nu \leq n _ \nu$) or $\cos k _ \nu t _ \nu$( $0 \leq k _ \nu \leq n _ \nu$). Multi-dimensional splines have a wide field of applications; they consist of "pieces" of algebraic polynomials in $m$ variables "tied" together in order to obtain specified smoothness properties. In the case $m = 2$, splines of the simplest form are formed by polynomials "tied" together along straight lines parallel to the coordinate axes. As a tool of approximation functions $g ( t _ {1} \dots t _ {m} )$ are used which are polynomials or splines only in one of the variables. For the approximation of non-periodic functions given on the whole space $\mathbf R ^ {m}$( or on an unbounded subset of $\mathbf R ^ {m}$) one may use entire functions of exponential type. These can be represented in the form of sums of absolutely convergent power series

$$\tag{2 } G _ {n _ {1} \dots n _ {m} } ( t _ {i} \dots t _ {m} ) = \ \sum _ {\begin{array}{c} k _ \nu \geq 0 , \\ \nu = 1 \dots m \end{array} } a _ {k _ {1} \dots k _ {m} } t _ {1} ^ {k _ {1} } \dots t _ {m} ^ {k _ {m} }$$

under the condition that for any $\epsilon > 0$ and all complex $t _ {1} \dots t _ {m}$ one has

$$| G _ {n _ {1} \dots n _ {m} } ( t _ {1} \dots t _ {m} ) | \leq M _ \epsilon \ \mathop{\rm exp} \ \sum _ {\nu = 1 } ^ { m } ( n _ \nu + \epsilon ) | t _ \nu | ,$$

where the constant $M _ \epsilon$ depends only on $\epsilon$( see [1]). It should be noted that, unlike polynomials, the function (2) is determined by an infinite number of parameters.

In the multi-dimensional case one has Weierstrass' theorem on the possibility of approximating a function $f \in C (Q)$, i.e. continuous on a bounded and closed set $Q \subset \mathbf R ^ {m}$, or a function $f \in \widetilde{C} ( \mathbf R ^ {m} )$, continuous on the whole space $\mathbf R ^ {m}$ with period $2 \pi$ in each variable, with arbitrary accuracy by means of algebraic (respectively, trigonometric) polynomials. A similar result holds in the spaces $L _ {p} (Q)$ and $\widetilde{L} _ {p} ( \mathbf R ^ {m} )$( in the periodic case) ( $1 \leq p < \infty$). General results and theorems concerning properties of the best approximation, the existence and uniqueness, the characteristic properties of functions of best approximation, and general relations of duality when approximating by (means of) a convex set (of functions) and, in particular, by a subspace, can be extended to normed linear spaces of functions of $m$ variables (see [3] and [4]). However, to obtain explicit formulations of these theorems, taking into account a specific metric and specific properties of the approximating subspace in the multi-dimensional case, involves great difficulties.

Quite thorough investigations were carried out on questions concerning the connection between the smoothness properties of a function of several variables and the rate of decrease of its best approximation by algebraic and trigonometric polynomials, as well as by entire functions.

Let $Q$ be an arbitrary open set in $\mathbf R ^ {m}$( in particular, $Q = \mathbf R ^ {m}$), let $\mathbf e$ be a unit vector in $\mathbf R ^ {m}$, let $h > 0$ and let $Q _ {h \mathbf e }$ be the set of points $t \in Q$ such that the segment $[ t , t + h \mathbf e ] \in Q$. If $f \in L _ {p} (Q)$ and $1 \leq p \leq \infty$, then the quantity

$$\omega _ {\mathbf e } ( f ; \delta ) _ {L _ {p} (Q) } = \ \sup _ {h \leq \delta } \ \| f ( t + h \mathbf e ) - f (t) \| _ {L _ {p} ( Q _ {h \mathbf e } ) }$$

is called the modulus of continuity of the function $f ( t _ {1} \dots t _ {m} )$ in the direction $\mathbf e$ with respect to the metric of $L _ {p} (Q)$. The quantity

$$\omega ( f ; \delta ) _ {L _ {p} (Q) } = \ \sup _ {\mathbf e } \ \omega _ {\mathbf e } ( f ; \delta ) _ {L _ {p} (Q) }$$

is called the modulus of continuity of $f$ in $L _ {p} (Q)$.

In the periodic case, the best approximation $\widetilde{E} _ {n _ {1} \dots n _ {m} } ( f _ {\widetilde{L} _ {p} } ( \mathbf R ^ {m} ) )$ of a function $f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} )$ having (Sobolev generalized) partial derivatives

$$D _ \nu ^ {r _ \nu } f = \ \frac{\partial ^ {r _ \nu } }{\partial t _ \nu ^ {r _ \nu } } f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} )$$

(where the $r _ \nu \geq 0$ are integers and $D ^ {0} f = f$, $\nu = 1 \dots m$) by trigonometric polynomials $T _ {n _ {1} \dots n _ {m} }$, satisfies the inequality

$$\tag{3 } \widetilde{E} _ {n _ {1} \dots n _ {m} } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq M \sum _ {\nu = 1 } ^ { m } n _ \nu ^ {- r _ \nu } \omega _ {\mathbf e _ \nu } ( D _ \nu ^ {r _ \nu } f ; n _ \nu ^ {-1} ) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } ,$$

where $\mathbf e _ \nu$ is the unit vector directed along $t _ \nu$, and where the constant $M$ depends neither on $f$ nor on $n _ \nu$. For a function $f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} )$ having generalized partial derivatives

$$D ^ {\mathbf r } f = \ \frac{\partial ^ {r} }{\partial t _ {1} ^ {r _ {1} } \dots \partial t _ {m} ^ {r _ {m} } } f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} )$$

of order $r = r _ {1} + \dots + r _ {m}$( where $\mathbf r = ( r _ {1} \dots r _ {m} )$), the following inequalities hold:

$$\tag{4 } \widetilde{E} _ {n \dots n } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq \ \frac{M}{n ^ {r} } \sum _ {r _ {1} + \dots + r _ {m} =r } \omega ( D ^ {\mathbf r } f ; \ n ^ {-1} ) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } .$$

If

$$\omega ( D ^ {\mathbf r } f ; \delta ) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq K \delta ^ \alpha ,\ \ 0 < \alpha < 1 ,$$

i.e. if the function $D ^ {\mathbf r } f$ satisfies a Hölder condition, then

$$\tag{5 } \widetilde{E} _ {n \dots n } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq \ \frac{M}{n ^ {r + \alpha } } ,\ \ r = 0 , 1 ,\dots; \ \ 0 < \alpha \leq 1 .$$

In the last case the inverse theorem asserts that if for a function $f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} )$ inequality (5) holds for all $n = 1 , 2 \dots$ then the derivatives $D ^ {\mathbf r } f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} )$ exist and satisfy for any $\mathbf h \in \mathbf R ^ {m}$ the inequalities

$$\tag{6 } \| D ^ {\mathbf r } f ( \mathbf t + \mathbf h ) - D ^ {\mathbf r } f ( \mathbf t ) \| _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq \ K | \mathbf h | ^ \alpha$$

for $0 < \alpha < 1$; whereas

$$\tag{7 } \| D ^ {\mathbf r } f ( \mathbf t + \mathbf h ) - 2 D ^ {\mathbf r} f ( \mathbf t) - D ^ {\mathbf r } f ( \mathbf t - \mathbf h ) \| _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq K | \mathbf h |$$

for $\alpha = 1$. Here $K$ does not depend on the length $| \mathbf h | = ( h _ {1} ^ {2} + \dots + h _ {m} ) ^ {1/2}$ of the vector $\mathbf h = ( h _ {1} \dots h _ {m} )$.

Theorems analogous to those given above also hold for non-periodic functions $f \in L _ {p} ( \mathbf R ^ {m} )$ if entire functions of exponential type are used as tools of approximation. The above-mentioned results can be extended also to classes of functions whose smoothness is described in terms of moduli of continuity (moduli of smoothness) of a higher order (see [1]).

In the case of approximating functions $f \in L _ {p} (Q)$ by algebraic polynomials $P _ {n _ {1} \dots n _ {m} }$ on a bounded parallelepiped (as well on some other bounded sets) direct theorems similar to (3), (4) and (5) have been proved. Inverses of these theorems, like in the case of functions defined on a bounded interval, are possible only on a proper subset $Q _ {1}$ of $Q$. Inverse theorems are known assuming a better order of approximation in the neighbourhood of boundary of $Q$( see [13]); there are also direct theorems asserting the possibility of achieving such an improvement in a neighbourhood of angular points (see [14]). Necessary and sufficient conditions for a function $f$ to belong to a class $H _ {C} ^ {r + \alpha } (Q)$( defined in the metric of $C (Q)$) by conditions similar to the conditions (6) and (7)) at the cost of raising the order of approximation in a neighbourhood of the boundary (like in the one-dimensional case) are unknown (1983). However, one has the following results of a negative character (see [13]). Let $Q = \{ {\mathbf t } : {\mathbf t \in \mathbf R ^ {2} , | \mathbf t | \leq 1 } \}$. There does not exist a sequence $\lambda _ {n} ^ {( \alpha ) } ( | \mathbf t | )$, $n = 1 , 2 ,\dots$; $0 < \alpha < 1$, of functions in $Q$ having the two following properties:

1) For any function $f \in H _ {C} ^ \alpha (Q)$ there are a constant $M$ and a sequence of polynomials

$$P _ {n} ( \mathbf t ) = \ \sum _ {0 \leq k _ {1} + k _ {2} \leq n } a _ {k _ {1} , k _ {2} } ^ {(n)} t _ {1} ^ {k _ {1} } t _ {2} ^ {k _ {2} } ,\ \ n = 1 , 2 \dots$$

such that

$$\tag{8 } | f ( \mathbf t ) - P _ {n} ( \mathbf t ) | \leq M \lambda _ {n} ^ {( \alpha ) } ( | \mathbf t | ) ,\ \ \mathbf t \in Q .$$

2) The existence of a constant $M > 0$ and of a sequence of polynomials $P _ {n} ( \mathbf t )$ satisfying (8) implies that $f \in H _ {C} ^ \alpha (Q)$ for any function $f$ defined on $Q$.

To illustrate the specific nature of approximation of functions of several variables, one can mention the following results.

Let $\widetilde{E} _ {n _ {1} , n _ {2} } (f) _ {\widetilde{X} }$ be the best approximation of a $2 \pi$- periodic function $f$ of two variables by trigonometric polynomials $T _ {n _ {1} , n _ {2} }$ in the metric of $\widetilde{X}$( $\widetilde{X} = \widetilde{C} ( \mathbf R ^ {2} )$ or $\widetilde{X} = \widetilde{L} _ {p} ( \mathbf R ^ {2} )$), and let $E _ {n _ {1} , \infty } (f) _ {\widetilde{X} }$ be the best approximation of $f$ in $\widetilde{X}$ by functions $T _ {n _ {1} , \infty }$, i.e. by trigonometric polynomials of degree at most $n _ {1}$ in the variable $t _ {1}$ and with coefficients that are functions of $t _ {2}$. The quantity $\widetilde{E} _ {\infty , n _ {2} } (f) _ {\widetilde{X} }$ is defined similarly.

If $1 < p < \infty$, one has the inequalities

$$\widetilde{E} _ {n _ {1} , n _ {2} } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {2} ) } \leq \ A _ {p} \{ \widetilde{E} _ {n _ {1} , \infty } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {2} ) } + E _ {\infty , n _ {2} } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {2} ) } \} ,$$

where $A _ {p}$ depends only on $p$.

If $\widetilde{X} = \widetilde{L} _ {1} ( \mathbf R ^ {2} )$ or $\widetilde{X} = \widetilde{C} ( \mathbf R ^ {2} )$, then

$$\tag{9 } \widetilde{E} _ {n _ {1} , n _ {2} } (f) _ {\widetilde{X} } \leq$$

$$\leq \ A \mathop{\rm ln} ( 2 + \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) ( \widetilde{E} _ {n _ {1} , \infty } (f) _ {\widetilde{X} } + \widetilde{E} _ {\infty , n _ {2} } (f) _ {\widetilde{X} } ) ,$$

where $A$ is an absolute constant, and the factor $\mathop{\rm ln} ( 2 + \mathop{\rm min} \{ n _ {1} , n _ {2} \} )$ in (9) cannot be replaced by another factor tending slower to infinity as $\mathop{\rm min} \{ n _ {1} , n _ {2} \} \rightarrow \infty$( see [15]).

Fundamental peculiarities arise in problems of interpolation for functions of several variables. For instance, contrary to the one-dimensional case, the existence of an algebraic interpolation polynomial depends substantially on the position of the interpolation nodes. Nevertheless, effective methods of constructing polynomials and splines interpolating a function $f ( t _ {1} \dots t _ {m} )$ on a distribution of nodes chosen in a definite way, have been established (see Interpolation). For multi-dimensional interpolation by splines, in a number of cases order estimates of the approximation errors, for the function $f$ as well as for its partial derivatives, have been found; in this direction, two-dimensional splines of low degree, as well as local (Hermitian) splines of arbitrary degree, have been studied in more detail (see [7], [10][12]). Among the other linear methods for approximating functions of several variables, multiple Fourier sums and their various means have been studied most extensively. Here, order estimates of the approximation with respect to classes of functions are known, and in some case asymptotically exact results were obtained (see [5], [6] and [8]).

#### References

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