Table Of ContentDie Grundlehren der
mathematischen Wissenschaften
in Einzeldarstellung
mit besonderer Berucksichtigung
der Anwendungsgebiete
Band 205
Herausgegeben von
J. J.
S. S. Chern L. Doob Douglas, jr.
A. Grothendieck E. Heinz F. Hirzebruch
E. Hopf W. Maak S. MacLane
W. Magnus M. M. Postnikov F. K. Schmidt
W. Schmidt D. S. Scott K. Stein
Geschaftsjiihrende Herausgeber
J.
B. E<;:kmann K. Moser B. L. van der Waerden
S. M. Nikol'skii
Approximation of Functions
of Several Variables
and Imbedding Theorems
Translated from the Russian by
J.
M. Danskin
Springer -Verlag
Berlin Heidelberg New York 1975
Sergei Mihauovic Nikol'skii
Steklov Mathematical Institute
Academy of Sciences, Moscow, U.S.S.R
Translator:
John M. Danskin (U.S.A.)
Title of the Russian Original Edition:
Priblizenie Funkci'i. Mnogih
Peremennyh i Teoremy Vlozeniya
Publisher: "Nauka", Moscow 1969
AlVIS Subject Classifications (1970)
33 A 10, 35 C 10, 40 B 05, 41 A 50, 41 A 60, 42 A 24,
42 A 68, 44 A 40, 46 E 35, 46 E 25, 46 F 05, 46 E 30
ISBN- 13: 978-3-642-65713-9 e-lSBN -13 :978-3-642-65711-5
DOl: 10.1007/978-3-642-65711-5
Library of Congress Catalog Card Number 74·4652.
This work is subject to copyright. All rights are reserved, whether the whole or part of the materjal is
concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by
photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright
Law where copies are made fore other than private usc, a fec is payable to the publisher, the amount of the
fee to be determined by agreement with the publisher.
© by Springer-Verlag. Berlin. Heidelberg 1975.
Softcover reprint of the hardcover 1st edition 1975
Author's Preface to the English Edition
This English translation of my book "PribliZenie Funkcir Mnogih
Peremennyh i Teoremy Vlozel1iya" is identical in content with the Rus
sian original, published by "Nauka" in 1969. However, I have corrected
a number of errors.
I am grateful to the publishing house Springer-Verlag for making
my book available to mathematicians who do not know Russian.
I am also especially grateful to the translator, Professor John M. Dan
skin, who has fulfilled his task with painstaking care. In doing so he has
showed high qualifications both as a mathematician and as a translator
of Russian, which is considered by many to be a very difficult language.
The discussion in this book is restricted, for the most part, to func
tions everywhere defined in n-dimensional space. The study of these
questions for functions given on bounded regions requires new methods.
In. connection with this I note that a new book, "Integral Represen
tations of Functions and Imbedding Theorems", by O. V. Besov,
V. P. Il'in, and myself, has just (May 1975) been published, by the
publishing house "Nauka", in Moscow.
Moscow, U.S.S.R., May 1975
S. M. Nikol'skir
Translator's Note
I am very grateful to Professor Nikol'skir, whose knowledge of English,
which is considered by many to be a very difficult language, is excellent,
for much help in achieving a correct translation of his book. And I join
Professor Nikol'skir in thanking Springer-Verlag. The editing problem
was considerable, and the typographical problem formidable.
Regensburg, West Germany
May 1975
John lVI. Danskin, jr.
Contents
Introduction 1
Chapter 1. Preparatory Information 5
1.1. The Spaces C(l$) and Lp{l$) 5
1.2. Normed Linear Spaces . . 9
1.3. Properties of the Space Lp{l$) . 19
1.4. Averaging of Functions According to Sobolev 27
1.5. Generalized Functions . . . . . 30
Chapter 2. Trigonometric Polynomials . 81
2.1. Theorems on Zeros. Linear Independence 81
2.2. Important Examples of Trigonometric Polynomials 84
2.3. The Trigonometric Interpolation Polynomial of Lagrange. 88
2.4. The Interpolation Formula of M. Riesz . . . . 90
2.5. The Bernstein's Inequality . . . . . . . . . . . . . 92
2.6. Trigonometric Polynomials of Several Variables . . . . 93
2.7. Trigonometric Polynomials Relative to Certain Variables 95
Chapter 3. Entire Functions of Exponential Type, Bounded on lR". 98
3.1. Preparatory Material. . . . . . . . . . . . . . . . . . 98
3.2. Interpolation Formula . . . . . . . . . . . . . . . . . 111
3.3. Inequalities of Different Metrics for Entire Functions of Expo-
nential Type ..................... 122
3.4. Inequalities of Different Dimensions for Entire Functions of
Exponential Type . . . . . . . . . . . . : . . .. 131
3.5. Subspaces of Functions of Given Exponential Type . .. 134
3.6. Convolutions with Entire Functions of Exponential Type 135
Chapter 4. The Function Classes W, H, B . . . 141
4.1. The Generalized Derivative . . . . . . . 141
4.2. Finite Differences and Moduli of Continuity 146
4.3. The Classes W, H, B . . . . . . . . . . 152
4.4. Representation of an Intermediate Derivate in Terms of a
Derivative of Higher Order and the Function. Corollaries 163
Contents VII
4.5. More on Sobolev Averages . . . . . . . . . . 174
4.6. Estimate of the Increment Relative to a Direction. 176
4.7. Completeness of the Spaces W, H, B . . . . . . 177
4.8. Estimates of the Derivative by the Difference Quotient 180
Chapter 5. Direct and Inverse Theorems of the Theory of Appro-
ximation. Equivalent Norms . 183
5.1. Introduction . . . . . 183
5.2. Approximation Theorem 185
5.3. Periodic Classes . . . . 193
5.4. Inverse Theorems of the Theory of Approximations 200
5.5. Direct and Inverse Theorems on Best Approximations. Equi-
valent H-Norms . . . . . . . . . . . . . . . . . . . . 207
5.6. Definition of B-Classes with the Aid of Best Approximations.
Equivalent Norms ................... 217
Chapter 6. Imbedding Theorems for Different Metrics and Dimen-
slons ........... ; ...... . 231
6.1. Introduction . . . . . . . . . . . . 231
6.2. Connections among the Classes B, H, W . 234
6.3. Imbedding of Different Metrics 236
6.4. Trace of a Function . . . . . . . . . 238
6.5. Imbeddings of Different Dimensions . . 240
6.6. The Simplest Inverse Theorem on Imbedding of Different
Dimensions . . . . . . . . . . . . . . . . . . . 244
6.7. General Imbedding Theorem for Different Dimensions. . . . 247
6.8. General Inverse Imbedding Theorem . . . . . . . . . . . 248
6.9. Generalization of the Imbedding Theorem for Different
Metrics. . . . . . . . . . 252
6.10. Supplementary Information ............... 255
Chapter 7. Transitivity and Unimprovability of Imbedding Theo
rems. Compactness. . . . . . . . . . . . . . . . . . . . . . 261
7.1. Transitive Properties of Imbedding Theorems . . . . . . . 261
7.2. Inequalities with a Parameter e. Multiplicative Inequalities 263
7.3. Boundary Functions in H;. Unimprovability of Imbedding
Theorems. . . . . . . . . . . . . . . . . . . . 267
7.4. More on Boundary Functions in H;. . . . . . . . . . . . 270
7.5. Unimprovability of Inequalities for Mixed Derivatives . . . 273
7.6. Another Proof of the Unimprovability of Imbedding Theo-
rems . . . . . . . . . . 274
7.7. Theorems on Compactness. . . . . . . . . . . . . . . . 279
VIII Contents
Chapter 8. Integral Representations and Isomorphism of Isotropy
Classes. . . . . . . . . . . . . . 289
8.1. The Bessel-MacDonald Kernels. . . . . . 289
8.2. The Isomorphism Classes W~. . .. . . . 294
8.3. Properties of the Bessel-MacDonald Kernel 296
rt
8.4. Estimate of the Best Approximation for I 298
8.5. Multipliers Equal to Unity on a Region. . 300
8.6. de la Vallee Poussin Sums of a Regular Function 301
8.7. An IneqUality for the Operation L, (r > 0) over Functions of
Exponential Type . . . . . . . . . . . . . . . . . . . 308
8.8. Decomposition of a Regular Function into Series Relative to de
la Vallee Poussin Sums . . . . . . . . . . . . . . . . . 312
8.9. Representation of Functions of the Classes ~8 in Terms of de
la Vallee Poussin Series. Null Classes (1 ~ P ~ 00) 313
8.10. Series Relative to Dirichlet Sums (1 < P < 00). 316
Chapter 9. The Liouville Classes L . . 323
9·1. Introduction . . . . . . . . . . . . . . . 323
9.2. Definitions and Basic Properties of the Classes L~ and L~ 325
9.3. Interrelationships among Liouville and other Classes. 332
9.4. Integral Representation of Anisotropic Classes. 335
9.5. Imbedding Theorems. . . . . . . . . . . . 358
9.6. Imbedding Theorem with a Limiting Exponent 368
9.7. Nonequivalence of the Classes B~ and L~ 373
Remarks. 377
Literature 403
Index of Names . 415
Subject Index . 417
Introduction
In the last two decades the theory of imbeddings of classes of differen
tiable functions of several variables, whose origins are to be found as far
back as the work of Sobolev in the 1930's, has been considerably devel
oped. At present a number of fundamental questions have been brought
to completion, and the need for a compact exposition has become appar
ent. As to myself, I came to the questions of the theory of imbedding in
connection with ideas long of interest to me connected with the classical
theory of approximation of functions by polynomials, first of all by
trigonometrical polynomials and their nonperiodic analogues, entire
functions of exponential type.
These ideas, which I had to develop within the new context, served
as a basis for me in the construction of the theory of imbeddings of
H-classes. Here, already in questions on the traces of functions, there
appeared not only direct theorems, but theorems completely inverting
them. We may give the name "inverse theorems" to theorems on the
extension of functions to a space from manifolds of lower dimensionality
lying in it. Here we include not only the isotropic case of functions
having differential properties which are the same in various directions,
but also the anisotropic case.
Later on O. V. Besov constructed an analogous theory of imbedding
of the B-classes introduced by him, also basing his work on the methods
of the theory of approximation by trigonometric polynomials or by
entire functions of exponential type. B-classes are significant because
they, like the H-classes, are, as we say, closed in themselves relative
to the imbedding theorems. We mean that the imbedding theorems of
interest to us (we will not formulate them here) are expressed in terms
of B-classes and in addition possess in a certain sense properties of
transitivity and invertibility in the case of the problem of traces.
Sobolev proved his imbedding theorems for the classes W~ = W~(.Q)
of functions having, on a sufficiently general region .Q of n-dimensional
space lR", derivatives to order l inclusive, which are integrable to the pth
power (1 ~ P ~oo). The Sobolev classes may be called "discrete classes",
because the parameter l expressing the differential properties of the
functions entering into them runs through the discrete sequence
l = 0, 1, 2, .... In this sense the classes Hand B are continuous. The
reader on familiarizing himself with Chapter 9 will realize that those
2 Introduction
theorems of Sobolev, with the complements to them due to V. 1. Kon
drasov and V. P. Il'in which are accompanied by a change in the metric,
are in a certain sense final, and even, to the extent that they make
discreteness of classes possible, transitive.
As to imbedding theorems accompanied only by a change in the
dimension without a change in the metric, theorems which we call
theorems on traces, the matter is more complicated. Of course, the
theorems of Sobolev gave a certain answer to the question as to what
differential properties were possessed by the trace of a function of class
r
W~(.Q) on a manifold c.Q, but this answer was given in the terms of
the classes W, and now we know that generally speaking if P =1= 2 the
final answer to this question is not expressed in terms of the classes W.
The first final results on the problem of traces of W-classes were
obtained for p = 2 by Aronszajn [1J, and independently by V. M. Babic
and L. N. Slobodeckii [2]. In this case (P = 2) the fractional classes
WH.Q) and W~(r), corresponding to any positive real parameter l, were
introduced. In terms of these classes direct and completely inverse
imbedding theorems were obtained. In the notation adopted in this book,
W~ = L~ = B~. The later investigations of Gagliardo [1J, Besov [1, 2J,
P. 1. Lizorkin [9J and S. V. Uspenskii [1, 2J, led to a complete solution
of the problem of traces of functions of the classes W~ for any finite
p > 1. The reader will find in the same Chapter 9 how this solution looks
(putting W~ = L~). Now we shall only say that the traces of functions
t of class W~ for P =1= 2 belong generally speaking not to W-classes but
to B-classes. This bears on the one hand on the fact that theorems on
imbedding of different dimensions (theorems on traces) cease to be closed
relative to the W-classes, and on the other hand on the fact that there
is a close connection between the classes Wand B. This connection was
so close that in the days when not everything concerning these questions
was clear, it was believed that the classes B~ for fractionall were the
natural extensions of the integer classes of Sobolev, and they were denoted
by W~. In fact, the natural such extensions are the so-called Liouville
classes L~, to which Chapter 9 is devoted. Thus the classes Ware treated
there as well, since we take W~ = L~ (l = 0, 1, ...) . The reader should
bear in mind that in this book the notation W~ is only used for l = 0,1, ....
In this connection see 4.3.
Sobolev studied the functions of his classes using the integral repre
sentations introduced by himself. A considerable development of this
work was carried out by V. P. Il'in, and then by O. V. Besov; in this
connection see 6.10 below. Functions of the classes L~ are defined on
the entire space, and in their integral representations it is extremely
important to take care that the kernel of the representation decrease
to zero sufficiently rapidly at infinity. The well-known kernels of Bessel-
Introduction 3
MacDonald are of this sort. They are taken as basic for the repesentation
L;.
of functions of the class We say "as basic", since we are in fact
L;.
considering here anisotropic classes The kernels of their integral
representations are defined by complicating the MacDonald kernels.
In writing Chapter 9 I have made essential use bf materials given to
me by my colleague P. I. Lizorkin, who quite recently found an entire
L;,
system of imbedding theorems for general anisotropic classes 'where r
is any positive vector. His results are published for the present in the
form of a short note.
In the one-dimensional case, where the problem of traces does not
L;
arise, theorems for imbedding different metrics for the classes and,
H;,
for noninteger r, for the classes were already obtained in the papers
of Hardy and Littlewood.
The operations 1,,., defined by the Bessel-MacDonald kernels, are of
universal character. In this book they are studied and applied in various
situations. We use rather extensively the concept of generalized func
tion. Hence the book contains a short section where there is presented,
with complete proofs, only that information from the theory of gener
alized functions which the reader needs to know for the understanding
of what follows. I introduce the concept of a generalized function
regular in the sense of Lp, using the operation 1,,.. For regular functions
various proofs connected with multiplication by a generalized function
are greatly simplified. I use this fact widely, since the generalized func
tions appearing in this book are regular.
The operation 1,,. has also been applied in an interesting way in
Chapter 8. It realizes isomorphisms not only of the L-classes, but of the
B- and H -classes as well, and it can serve as a means for integral repre
sentations of functions of these classes. These ideas, which in the periodic
one-dimensional case go back to Hardy and Littlewood, were quite
recently studied from various points of view in the papers of Aronszajn
and Smith, Calderon, Taibleson, Lions, Lizorkin, the author and others.
It is natural that we have found in this book a place for the foun
dations of the theory of approximation of functions of several variables
by trigonometric polynomials and entire functions of exponential type.
They are of interest in themselves, but basically their role is subordinate.
We further prove, by means of the methods of the theory of approxima
tion, imbedding theorems for H- and B-classes, and give representations
of functions of these classes in terms of series in entire functions of
exponential type or in terms of trigonometric polynomials. Having these
objectives in mind, along with the traditional inequalities we introduce
and apply other inequalities, referring to different dimensions and
metrics.
It should be noted that we give in this book complete proofs of the
