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ΠΠΎΠ½ΡΡΡΡΠΊΡΠΈΠ²Π½ΡΠ΅ ΡΠ°Π·ΡΠ΅ΠΆΠ΅Π½Π½ΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΡ Π΄Π»Ρ ΡΡΠ½ΠΊΡΠΈΠΉ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½ΠΎΠΉ ΡΠΌΠ΅ΡΠ°Π½Π½ΠΎΠΉ Π³Π»Π°Π΄ΠΊΠΎΡΡΠΈ
ΠΠΎΠ»ΡΡΠ΅Π½Ρ ΠΊΠ°ΠΊ ΠΏΠΎΡΡΠ΄ΠΊΠΎΠ²ΡΠ΅ ΠΎΡΠ΅Π½ΠΊΠΈ (Π² ΡΠ»ΡΡΠ°Π΅ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΡ Π² ΡΠ°Π²Π½ΠΎΠΌΠ΅ΡΠ½ΠΎΠΉ ΠΌΠ΅ΡΡΠΈΠΊΠ΅), ΡΠ°ΠΊ ΠΈ ΡΠΎΡΠ½ΡΠ΅ ΠΏΠΎ ΠΏΠΎΡΡΠ΄ΠΊΡ ΠΎΡΠ΅Π½ΠΊΠΈ (Π² ΡΠ»ΡΡΠ°Π΅ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΡ Π² ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΠΎΠΉ ΠΌΠ΅ΡΡΠΈΠΊΠ΅) Π΄Π»Ρ Π½Π°ΠΈΠ»ΡΡΡΠ΅Π³ΠΎ m-ΡΠ»Π΅Π½Π½ΠΎΠ³ΠΎ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΡ ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΡΠ½ΠΊΡΠΈΠΉ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½ΠΎΠΉ ΡΠΌΠ΅ΡΠ°Π½Π½ΠΎΠΉ Π³Π»Π°Π΄ΠΊΠΎΡΡΠΈ ΠΈΠ· ΠΊΠ»Π°ΡΡΠΎΠ², Π±Π»ΠΈΠ·ΠΊΠΈΡ
ΠΊΠ»Π°ΡΡΠ°ΠΌ ΡΠΈΠΏΠ° ΠΠΈΠΊΠΎΠ»ΡΡΠΊΠΎΠ³ΠΎβΠΠ΅ΡΠΎΠ²Π°. ΠΡΠΈ ΡΡΠΎΠΌ ΠΊΠ°ΠΆΠ΄Π°Ρ ΠΈΠ· Π²Π΅ΡΡ
Π½ΠΈΡ
ΠΎΡΠ΅Π½ΠΎΠΊ ΡΠ΅Π°Π»ΠΈΠ·ΡΠ΅ΡΡΡ ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΈΠ²Π½ΡΠΌ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠΌ, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΡΠΌ Π½Π° ΠΆΠ°Π΄Π½ΡΡ
Π°Π»Π³ΠΎΡΠΈΡΠΌΠ°Ρ
.The order bounds (in the case of uniform metrics) and exact order bounds (in the case of integral metrics) for the best m-term trigonometric approximation of periodic functions with generalized mixed smoothness from classes close to the NikolβskiiβBesov-type ones are obtained. Every of the upper bounds is realized by a constructive method based on greedy algorithms
Nonlinear tensor product approximation of functions
We are interested in approximation of a multivariate function
by linear combinations of products
of univariate functions , . In the case it is a
classical problem of bilinear approximation. In the case of approximation in
the space the bilinear approximation problem is closely related to the
problem of singular value decomposition (also called Schmidt expansion) of the
corresponding integral operator with the kernel . There are known
results on the rate of decay of errors of best bilinear approximation in
under different smoothness assumptions on . The problem of multilinear
approximation (nonlinear tensor product approximation) in the case is
more difficult and much less studied than the bilinear approximation problem.
We will present results on best multilinear approximation in under mixed
smoothness assumption on
Approximation algorithms for wavelet transform coding of data streams
This paper addresses the problem of finding a B-term wavelet representation
of a given discrete function whose distance from f is
minimized. The problem is well understood when we seek to minimize the
Euclidean distance between f and its representation. The first known algorithms
for finding provably approximate representations minimizing general
distances (including ) under a wide variety of compactly supported
wavelet bases are presented in this paper. For the Haar basis, a polynomial
time approximation scheme is demonstrated. These algorithms are applicable in
the one-pass sublinear-space data stream model of computation. They generalize
naturally to multiple dimensions and weighted norms. A universal representation
that provides a provable approximation guarantee under all p-norms
simultaneously; and the first approximation algorithms for bit-budget versions
of the problem, known as adaptive quantization, are also presented. Further, it
is shown that the algorithms presented here can be used to select a basis from
a tree-structured dictionary of bases and find a B-term representation of the
given function that provably approximates its best dictionary-basis
representation.Comment: Added a universal representation that provides a provable
approximation guarantee under all p-norms simultaneousl
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