On infinitely smooth compactly supported almost-wavelets

Berkolaiko, M. Z.; Novikov, Igor

doi:10.1007/bf02362405

Cited by 11 publications

(5 citation statements)

References 2 publications

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“…The notion of nonstationary wavelet system is introduced independently by M. Z. Berkolayko, I. Y. Novikov [2] and by C. de Boor, R. DeVore, A. Ron [3]. In [2], the nonstationary system (called almost-wavelets) is used to construct an orthonormal shift invariant basis consisting of infinitely differentiable compactly supported functions. It is well known that it is impossible to construct stationary wavelet basis satisfying these properties.…”

Section: Introductionmentioning

confidence: 99%

“…The frameworks of nonstationary nonperiodic and periodic wavelets are introduced and studied separately. The notion of nonstationary wavelet system is introduced independently by M. Z. Berkolayko, I. Y. Novikov [2] and by C. de Boor, R. DeVore, A. Ron [3]. In [2], the nonstationary system (called almost-wavelets) is used to construct an orthonormal shift invariant basis consisting of infinitely differentiable compactly supported functions.…”

Section: Introductionmentioning

confidence: 99%

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On a connection between nonstationary and periodic wavelets

Lebedeva

2017

Journal of Mathematical Analysis and Applications

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We compare frameworks of nonstationary nonperiodic wavelets and periodic wavelets. We construct one system from another using periodization. There are infinitely many nonstationary systems corresponding to the same periodic wavelet. Under mild conditions on periodic scaling functions, among these nonstationary wavelet systems, we find a system such that its time-frequency localization is adjusted with an angular-frequency localization of an initial periodic wavelet system. Namely, we get the following equality lim j→∞ U C B (ψ P j ) = lim j→∞ U C H (ψ N j ), where U C B and U C H are the Breitenberger and the Heisenberg uncertainty constants, ψ P j ∈ L 2 (T) and ψ N j ∈ L 2 (R) are periodic and nonstationary wavelet functions respectively.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On a connection between nonstationary and periodic wavelets

Lebedeva

2017

Journal of Mathematical Analysis and Applications

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“…They described a wide class of functional-differential equations with compactly supported solutions, investigated functions up x ( ) and other atomic functions in detail, and noted the importance of the general approach to the construction of atomic functions for applications. M. Z. Berkolaiko and I. D. Novikov note that compactly supported infinitely smooth near-splashes [9,10] were constructed by them as a result of synthesis of ideas of I. Daubechies [56] and the theory of atomic functions.…”

mentioning

confidence: 98%

“…The paper considered the application of atomic functions in conjunction with the mathematical tools of the theory of R-functions in variational methods for solving problems of mathematical physics. In some works of foreign authors [9][10][11][12][13][14][15][16][17][18][19][20][21][22], the atomic function up x ( )also gained ground as "the Rvachev function." 893 1060-0396/07/4306-0893…”

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confidence: 99%

Atomic functions: Generalization to the multivariable case and promising applications

Kolodyazhny

Rvachov

2007

Cybern Syst Anal

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517.95Main trends in the theory of atomic functions are outlined. These functions have been studied for more than 35 years and gave rise to new research areas in mathematical analysis, approximation theory, numerical methods, etc. Atomic functions are infinitely differentiable solutions with compact support to special functional-differential equations.Atomic functions whose investigation for the case of one independent variable has received significant attention until quite recently are infinitely differentiable compactly supported solutions to functional-differential equations of the formwhere L is a linear differential operator with constant coefficients. Atomic functions have been intensively investigated for more than 35 years (see [1] and reviews in [2-7]), and some new scientific directions arise in which typical properties inherent in atomic functions are necessarily required in theoretical and practical developments. Atomic functions and spaces generated by them are used in mathematical analysis [2,4,6], approximation theory [4, 5], in solving differential equations and boundary problems of mathematical physics [2, 3, 11, 15-21, 44, 48, 80], in numerical methods and other fields of applied mathematics [22, 46-47, 59-62], and also in technical realizations of bell-shaped functions [30, 31], digital signal processing [34-37, 41, 50-52], analysis and synthesis of antennas [49-52], etc. The history of atomic functions begins with [1], which was published by V. L. Rvachev and V. A. Rvachev in 1971 and in which, in particular, they used the method of contracting mappings to prove the existence and uniqueness of a compactly supported solution of the equation y x y x y x ¢ = + --( ) ( ) ( ) 2 2 1 2 2 1 under the conditions y ( ) 0 1 = and supp y x ( ) [ , ] = -1 1 . This problem was stated by V. L. Rvachev as long ago as in 1967 at a seminar on problems of applied mathematics, and its solution was connected with the advent of the most important atomic function up x ( ), up x itx t t dt k k k ( ) exp ( ) sin ( ) = -¥ ¥ = ¥ --ò Õ 1 2 2 2 1 p .The term "atomic function" was introduced in [2] in 1975. The paper considered the application of atomic functions in conjunction with the mathematical tools of the theory of R-functions in variational methods for solving problems of mathematical physics. In some works of foreign authors [9][10][11][12][13][14][15][16][17][18][19][20][21][22], the atomic function up x ( )also gained ground as "the Rvachev function." 893 1060-0396/07/4306-0893

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“…Построенные и исследуемые автором n-раздельные всплески можно также назвать частным случаем введенных М. З. Берколайко и И. Я. Новиковым так называемых нестационарных, или почти всплесков [3]. В отличие от нестационарных всплесков, когда на каждом уровне j базис пространства V j образован сдвигами своей функции ϕ j , мы ограничиваемся конечным числом масштабирующих функций.…”

unclassified

Приближение функций $n$-раздельными всплесками в пространствах $\mathbf{L}^p(\mathbb{R})$, $1 \leq p \leq \infty$

Плещева¹

2019

Trudy IMM UrO RAN

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В работе рассматриваются построенные автором ранее ортонормированные базисы n-раздельных КМА и всплесков. В классическом случае базис пространства L 2 (R) образован сдвигами и сжатиями единственной функции ψ. В отличие от классического случая, в данной статье несколько базисов пространства L 2 (R) образованы сдвигами и сжатиями n функций ψ s , s = 1,. .. , n. Построенные n-раздельные всплески образуют ортонормированный базис пространства L 2 (R). В этом случае ряд n s=1 j∈Z k∈Z f, ψ s nj+s,k ψ s nj+s,k сходится к функции f в пространстве L 2 (R). Мы привели дополнительные ограничения на функции ϕ s и ψ s , s = 1,. .. , n, обеспечивающие сходимость такого ряда к функции f в пространствах L p (R), 1 ≤ p ≤ ∞ по норме и почти всюду. Ключевые слова: всплеск, масштабирующая функция, базис, кратномасштабный анализ. E. A. Pleshcheva. Approximation of functions by n-separate wavelets in the spaces L p (R), 1 ≤ p ≤ ∞. We consider the orthonormal bases of n-separate MRAs and wavelets constructed by the author earlier. The classical wavelet basis of the space L 2 (R) is formed by shifts and compressions of a single function ψ. In contrast to the classical case, we consider a basis of L 2 (R) formed by shifts and compressions of n functions ψ s , s = 1,. .. , n. The constructed n-separate wavelets form an orthonormal basis of L 2 (R). In this case, the series n s=1 j∈Z k∈Z f, ψ s nj+s ψ s nj+s converges to the function f in the space L 2 (R). We write additional constraints on the functions ϕ s and ψ s , s = 1,. .. , n, that provide the convergence of the series to the function f in the spaces L p (R), 1 ≤ p ≤ ∞, in the norm and almost everywhere.

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On infinitely smooth compactly supported almost-wavelets

Cited by 11 publications

References 2 publications

On a connection between nonstationary and periodic wavelets

On a connection between nonstationary and periodic wavelets

Atomic functions: Generalization to the multivariable case and promising applications

Приближение функций $n$-раздельными всплесками в пространствах $\mathbf{L}^p(\mathbb{R})$, $1 \leq p \leq \infty$

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