On discontinuous Sturm-Liouville problems with transmission conditions
|
|
- Claribel Alexander
- 5 years ago
- Views:
Transcription
1 J. Math. Kyoto Uiv. (JMKYAZ 44-4 (4, O discotiuous Sturm-Liouville problems with trasmissio coditios By O. Sh. Mukhtarov, MahirKadakal ad F. S. Muhtarov Abstract We cosider a discotiuous Sturm-Liouville equatio together with eigeparameter depedet boudary coditios ad two supplemetary trasmissio coditios at the poit of discotiuity. By modifyig some techiques of [], [] ad [4] we exted ad geeralize some approach ad results of classic regular Sturm-Liouville problems to the similar problems with discotiuities. I particular, we itroduce a special Hilbert space formulatio such a way that the cosidered problem ca be iterpreted as a eigevalue problem of suitable self-adjoit operator, the we costruct the Gree fuctio ad resolvet operator ad derive a asymptotic formulas for eigevalues ad ormalized eigefuctios.. Itroductio The Sturmia theory is a importat aid i solvig may problems of mathematical physics. Usually, the eigevalue parameter appear liearly oly i the differetial equatio of the classic Sturm-Liouville problems. However, i mathematical physics are ecoutered such problems, where eigevalue parameter appear i both differetial equatio ad boudary coditios (various physical applicatios ca be foud i []. There is a substatial literature o this type of problems (see, for example, [], [], [3], [8], [9], [4] ad more recetly [5], [6], [7] ad correspodig refereces cited therei. I these works, oly cotiuous problems have bee ivestigated. The purpose of this paper is to exted some classic results of Sturmia theory to the discotiuous case, i which two supplemetary trasmissio coditios added to the boudary coditios. I fact, we ivestigate both cotiuous ad discotiuous cases (the cases δ =adδ i below, respectively i this study. Let us cosider the Sturm-Liouville equatio (. τu := u + q(xu = λu for x [, (, ] 99 Mathematics Subject Classificatio(s. 34L, 34B4, 34B7 Received October, 3 Revised September, 4
2 78 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov (i.e. o [, ] except oe ier poit x =,whereq(x is a real-valued, cotiuous i both [, ad (, ] ad has fiite limites q(± = lim q(x, x ± together with stadard boudary coditio at x = (. L u := α u( + α u ( =, trasmissio coditios at the poit of discotiuity x = (.3 L u := u( δu(+ =, (.4 L 3 u := u ( δu (+ =, ad eigeparameter depedet boudary coditio at x = (.5 L 4 (λu := λ(β u( β u ( + (β u( β u ( =, where λ C is a complex spectral parameter ad all coefficiets of the boudary ad trasmissio coditios are real costats. Naturally, we assume that α + α,δ, β + β ad β + β. Moreover, we shall assume that ρ := β β β β >. Some special cases of this problem arises after a applicatio of the method of separatio of variables to the varied assortmet of physical problems, such as, i heat ad mass trasfer problems (see, for example, [], i vibratig strig problems whe the strig loaded additioally with poit masses (see, for example, [], i thermal coductio problem for a thi lamiated plate (see, for example, []. Note that such properties as isomorphism, coerciveess with respect to the spectral parameter, completeess of root fuctios, distributios of eigevalues of some discotiuous boudary value problems with trasmissio coditios ad its applicatios to the correspodig iitial-boudary value problems for parabolic equatios have bee ivestigated i [5], [6], [7], [].. Operator-theoretic formulatio i suitable Hilbert space I this sectio, we itroduce the special ier product i the Hilbert space (L (, L (, C ad defie a liear operator A i it such a way that the cosidered problem (. (.5 ca be iterpreted as the eigevalue problem of A. So, we defie a ew Hilbert space ier product o H := (L (, L (, C by for F = ( f(x f F, G H = δ f(xg(xdx + δ f(xg(xdx + δ ρ f g (, G = g(x H. For coveiece we shall use the otatios g R (u :=β u( β u (, R (u :=β u( β u (.
3 O discotiuous Sturm-Liouville problems with trasmissio coditios 78 I this Hilbert space, we costruct the operator A : H H with domai (. D(A = ad actio law (. AF = F = ( f(x f f(x,f (xare absolutely cotiuous i [, (, ] ad have fiite oe-had sided limits f(,f (, respectively; τf L (, L (, L f = L f = L 3 f =;f = R (f ( ( τf f(x with F = R (f R (f D(A. Thus, we ca pose the boudary value-trasmissio problem (. (.5 as ( u(x (.3 AU = λu, U := R (u D(A i the Hilbert space H. It is readily verified that the eigevalues of the operator A coicide with those of the problem (. (.5. Theorem.. The operator A is symmetric. Proof. Let F = ( f(x R (f ad G = ( g(x R (g are arbitrary elemet of D(A. By two partial itegratio we get (.4 AF, G H F, AG H = W (f,g; W (f,g; + δ W (f,g; δ δ δ W (f,g;++ δ ρ (R (fr (g R (fr (g, where, as usual, W (f,g; x deotes the Wroskias of the fuctios f ad g, i.e. W (f,g; x :=f(xg (x f (xg(x. Sice F, G D (A, the first compoets of these elemets, i.e. f ad g satisfy the boudary coditio (.. From this fact, we easily have that (.5 W (f,g; =, sice α ad α are real. Further, as f ad g also satisfy both trasmissio coditios we get (.6 W (f,g; = δ W (f,g;+.
4 78 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov Moreover, the direct calculatios gives (.7 R (fr (g R (fr (g = ρw (f,g;. Now, substitutig (.5 (.7 i (.4 gives AF, G H = F, AG H (F, G D(A, so A is symmetric. The proof is complete. Recallig that the eigevalues of the problem (. (.5 are coicide with the eigevalues of A we have the ext corollary. Corollary.. All eigevalues of the problem (. (.5 are real. As all eigevalues are real it is eough to ivestigate oly the real-valued eigefuctios. Takig this ito accout, we ca ow assume that all eigefuctios of the problem (. (.5 are real-valued. 3. Asymptotic represetatios of the basic solutios Let us defie two basic solutios { { φ (x, λ, x [, χ (x, λ, x [, φ(x, λ = ad χ(x, λ = φ (x, λ, x (, ] χ (x, λ, x (, ] of equatio (. by the followig procedure. At first cosider the ext iitial-value problem: (3. u + q(xu = λu, x [, ], (3. u( = α, (3.3 u ( = α. By virtue of [, Theorem.5] this problem has a uique solutio u = φ (x, λ,whichisaetirefuctioofλ C for each fixed x [, ]. Slightly modifyig the method of [, Theorem.5] we ca prove that the iitial-value problem (3.4 u + q(xu = λu, x [, ], (3.5 u( = β λ + β, (3.6 u ( = β λ + β has a uique solutio u = χ (x, λ, which is a etire fuctio of parameter λ for each fixed x [, ]. The other fuctios φ (x, λ adχ (x, λ cabe defied i terms of φ (x, λ adχ (x, λ, respectively. Applyig the method used i the proof of [3, Theorem ] we ca prove that the iitial-value problem (3.7 u + q(xu = λu, x [, ], (3.8 (3.9 u( = δ φ (,λ, u ( = δ φ (,λ
5 O discotiuous Sturm-Liouville problems with trasmissio coditios 783 has a uique solutio u = φ (x, λ,whichisaetirefuctioofλ for each fixed x [, ]. Similarly, the iitial-value problem (3. (3. (3. u + q(xu = λu, x [, ], u( = δχ (,λ, u ( = δχ (,λ also has a uique solutio u = χ (x, λ, which is a etire fuctio of λ for each fixed x [, ]. By virtue of (3. ad (3.3 the solutio φ(x, λ satisfies the first boudary coditio (.. Moreover, by virtue of (3.8 ad (3.9, φ(x, λ also satisfies both trasmissio coditios (.3 ad (.4. Similarly, by virtue of (3.5, (3.6, (3. ad (3. the other solutio χ(x, λ satisfies the secod boudary coditio (.5 ad both trasmissio coditios (.3 ad (.4. It is well-kow, from the ordiary liear differetial equatios theory, that each of the Wroskias ω (λ =W (φ (x, λ,χ (x, λ ad ω (λ =W (φ (x, λ,χ (x, λ are idepedet o x i [, ] ad [, ], respectively. Lemma 3.. The equality ω (λ =δ ω (λ holds for each λ C. Proof. Sice the above Wroskias are idepedet o x, the usig (3.8, (3.9, (3. ad (3. we have (3.3 ω (λ =φ (,λχ (,λ φ (,λχ (,λ =(δφ (,λ (δχ (,λ (δφ (,λ (δχ (,λ = δ ω (λ. Corollary 3.. The zeros of ω (λ ad ω (λ are coicide. Takig the Lemma 3. ito accout we deote both ω (λ adδ ω (λ by ω(λ. Recallig the defiitios of φ i (x, λadχ i (x, λ we coclude the ext corollary. Corollary 3.3. The fuctio ω(λ is a etire fuctio. Theorem 3.4. The eigevalues of the problem (. (.5 are coicide with the zeros of the fuctio ω(λ. Proof. Let ω(λ =. The W (φ (x, λ,χ (x, λ = for all x [, ]. Cosequetly, the fuctios φ (x, λ adχ (x, λ are liearly depedet, i.e. χ (x, λ =k φ (x, λ,x [, ]
6 784 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov for some k. By usig (3. ad (3.3, from this equality we have α χ(,λ +α χ (,λ =α χ (,λ +α χ (,λ = k (α φ (,λ +α φ (,λ = k (α α + α ( α =, so χ(x, λ satisfies the first boudary coditio (.. Recallig that the solutio χ(x, λ satisfies also the other boudary coditio (.5 ad both trasmissio coditios (.3 ad (.4, we coclude that χ(x, λ is a eigefuctio of the problem (. (.5, i.e. λ is a eigevalue. Thus, each zero of ω(λ is a eigevalue. Now let λ be a eigevalue ad u (x be a ay eigefuctio correspodig to this eigevalue. Suppose, if possible, that ω(λ. Whece W (φ (x, λ,χ (x, λ adw (φ (x, λ,χ (x, λ. So, by virtue of well-kow properties of Wroskias, it follows that each of the pairs φ (x, λ, χ (x, λ ad φ (x, λ, χ (x, λ are liearly idepedet. Therefore the solutio u (x of Equatio (. may be represeted i the form { c φ (x, λ +c χ (x, λ,x [,, u (x = c 3 φ (x, λ +c 4 χ (x, λ,x (, ], where at least oe of the costats c,c,c 3 ad c 4 is ot zero. Cosiderig the true equalities L ν (u (x = c L ν (φ (x, λ + c L ν (χ (x, λ (3.4 + c 3 L ν (φ (x, λ + c 4 L ν (χ (x, λ =, ν=,, 3, 4 as the homogeeous system of liear equatios of the variables c,c,c 3 ad c 4, ad takig (3.8, (3.9, (3. ad (3. ito accout it follows that the determiat of this system is equal to ω (λ φ (,λ χ (,λ δφ (,λ δχ (,λ φ (,λ χ (,λ δφ (,λ δχ (,λ = δ ω (λ ω(λ ω (λ = δ ω3 (λ ad therefore it is ot equal to zero by assumptio. Cosequetly, this homogeeous system of liear equatios has the oly trivial solutio (c,c,c 3,c 4 = (,,,. Thus we get cotradictio, which completes the proof. Theorem 3.5. Let λ = s, Im s = t. The, the followig asymptotic equalities hold as λ : ( I the case α φ (k (x, λ =α d k ( (3.5 cos[s(x +]+O e t (x+, dxk s k φ (k (x, λ =α d k ( (3.6 cos[s(x +]+O e t (x+, δ dxk s k
7 O discotiuous Sturm-Liouville problems with trasmissio coditios 785 for k =ad k =. ( I the case α = (3.7 (3.8 φ (k (x, λ = α s φ (k (x, λ = α δs d k ( si[s(x +]+O dxk ( d k si[s(x +]+O dxk e t (x+ s k e t (x+ s k for k =ad k =. Moreover, each of asymptotic equalities hold uiformly for x. Proof. The above asymptotic formulas for φ (x, λ have bee foud i [, Lemma.7]. But the similar formulas for the solutio φ (x, λ eed idividual cosideratios, sice this solutio is defied by the iitial coditios havig special o-stadard forms. The iitial-value problem (3.7, (3.8, (3.9 ca be trasformed ito a equivalet itegral equatio u(x =δ φ (,λcos λx + δ λ φ (,λsi λx, (3.9 + λ x si[ λ(x y]q(yu(ydy. Let α. Substitutig (3.5 i (3.9 we have (3. φ (x, λ = α δ cos λ(x ++ λ + O ( λ e t (x+. x si[ λ(x y]q(yφ (y, λdy Multiplyig by e t (x+ ad lettig F (x, λ =e t (x+ φ (x, λ, we have the ext asymptotic itegral equatio F (x, λ = α e t (x+ cos λ(x + δ + x si[ ( λ(x y]q(ye t (x y F (y, λdy + O λ. λ Lettig M(λ = max F (x, λ from the last equatio we derive that x [,] M(λ M ( α δ + λ
8 786 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov for some M >. Cosequetly, M(λ =O( as λ,so φ (x, λ =O(e t (x+ as λ. Substitutig this asymptotic equality i the itegral term of the (3. gives (3.6 for the case k =. The proof of (3.6 for the case k =cabe obtaied at oce by differetiatig (3.9 ad the followig the same procedure as i the case k =. The proof of (3.8 is similar to that of (3.6 ad hece omitted. Theorem 3.6. Let λ = s, Im s = t. The, the followig asymptotic formulas hold for the eigevalues of the boudary-value-trasmissio problem (. (.5: Case : β, α (3. s = π( + O ( Case : β, α = (3. s = ( π Case 3: β =, α (3.3 s = ( π Case 4: β =, α = + O + O (3.4 s = π + O (, (, (,. Proof. Let us cosider Case oly. Writig ω (λ =φ (x, λχ (x, λ φ (x, λχ (x, λ for x = ad the usig χ (,λ=β λ + β, χ (,λ=β λ + β as give by (3.5 ad (3.6, respectively, we have the followig represetatio for ω (λ: (3.5 ω (λ =(β λ + β φ (,λ (β λ + β φ (,λ. Now writig x = i (3.6 ad the substitutig i (3.5 we derive that (3.6 ω (λ =δ β α s 3 si( λ+o( s e t. By applyig well-kow Rouche s Theorem (which assert that if f(z ad g(z are aalytic iside ad o a closed cotour Γ, ad g(z < f(z o Γ, the f(z adf(z+g(z have the same umber zeros iside Γ, provided that
9 O discotiuous Sturm-Liouville problems with trasmissio coditios 787 each zeros are couted accordig to their multiplicity o a sufficietly large cotour, it follows that ω (λ has the same umber of zeros iside the cotour as the leadig term i (3.6. Hece, if λ <λ <λ <..., are the zeros of ω (λ ads = λ,wehave (3.7 s = π ( + δ, where δ < π 4, for sufficietly large. By substitutig (3.7 ito (3.6 we have ( δ = O, so the proof completes for Case. The proofs for the other cases are similar. The followig asymptotic formulas hold for the eige- Theorem 3.7. fuctios φ λ (x = of the problem (. (.5: Case : β, α ( α cos (3.8 φ λ (x = α δ cos Case : β, α = { φ (x, λ, x [,, φ (x, λ, x (, ] π( (x + ( π( (x + (3.9 φ λ (x ( ( α π( / si π = ( ( α δ π( / si π Case 3: β =, α ( α cos π (3.3 φ λ (x = α δ cos, x [,, x (, ] (x + (x +,x [,, x (, ] ( (x +, x [, ( π ( Case 4: β =, α = ( α π si (3.3 φ λ (x = α δ π si (x +, x (, ] π(x + ( π(x +, x [,, x (, ] + O (, ( + O, + O (, ( + O.
10 788 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov All this asymptotic formulas hold uiformly for x. Proof. Let us cosider Case oly. Substitutig (3.6 ito the itegral term of (3., it is easy to see that (3.3 x si[ λ(x y]q(yφ (y, λdy = O(e t (x+. Substitutig ito (3. we have (3.33 φ (x, λ = α δ cos ( λ(x ++O λ e t (x+. We already kow that all eigevalues are real. Further, puttig λ = R, R> i (3.6 it follows that ω( R as R +, soω( R for sufficietly large R>. Cosequetly, the set of eigevalues is bouded below. Now, writig λ = s i (3.33 we obtai φ (x, λ = α ( δ cos[s (x +]+O sice t =Ims = for sufficietly large. After some routie calculatios we easily obtai that ( ( cos[s (x +]=cos π( (x + + O. Cosequetly, φ (x, λ = α δ cos Similarly we ca fid that Sice φ (x, λ =α cos φ λ (x = s ( π( (x + + O, (. ( ( π( (x + + O. { φ (x, λ,x [,, φ (x, λ,x (, ], the proof for Case is completed. The proofs for the other cases are similar. 4. Asymptotic formulas for ormalized eigefuctios It is evidet that the two-compoet vectors ( φλ (x (4. Φ := R (φ,=,,,... λ
11 O discotiuous Sturm-Liouville problems with trasmissio coditios 789 are the eigeelemets of the operator A correspodig to the eigevalue λ. For m, (4. Φ, Φ m H =,,m=,,,..., sice A is symmetric. Deotig (4.3 ψ := φ λ (x Φ H, it is easily see that the eigeelemets ( ψλ (x (4.4 Ψ := R (ψ,,m=,,,... λ are orthoormal. That is, where δ m is the kroecker delta. x = ad Ψ, Ψ m H = δ m, Lemma 4.. The followig asymptotic equalities hold: ( i case α (4.5 R (φ λ =O (, ( i case x = ( (4.6 R (φ λ =O. Proof. It follows from the equality ω (λ =that (4.7 λ R (φ λ +R (φ λ =. ( Let α. The from the formula (3.6 we get R (φ λ =β φ λ ( β φ λ ( = β O( β O( s. Now applyig Theorem 3.6 we have (4.8 R (φ λ =O(. Substitutig (4.8 ito (4.7, ad takig Theorem 3.6 ito accout, we get R (φ λ = ( R (φ λ =O. λ
12 79 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov Now let α =. By usig Theorem 3.6 we obtai R (φ λ =β φ λ ( β φ λ ( = β O( s β O( = β O = O(. ( β O( Takig ito accout that λ ( π ad usig (4.7 we have R (φ λ = ( R (φ λ =O λ. The proof is complete. Theorem 4.. Let Φ be defied as i (4.. The the followig asymptotic formulas hold for the orms Φ H of the eigeelemets Φ : Case : If β ad α,the (4.9 Φ H = α ( + O, δ Case : If β ad α =,the (4. Φ H = α ( δ π( / + O, Case 3: If β =ad α,the (4. Φ H = α ( + O, δ Case 4: If β =ad α =,the (4. Φ H = α ( δ π + O. Proof. Let β adα. I this case, usig (3.8 we have (4.3 (φ λ (x dx = α = α [ ( cos π( (x + + O cos ( π( (x + dx + O ( = α + O. ( ] dx (
13 O discotiuous Sturm-Liouville problems with trasmissio coditios 79 Similarly, we have (4.4 ( (φ λ (x dx = α δ + O. Usig (4.5, (4.3 ad (4.4 we get (4.5 Φ H = δ (φ λ (x dx + δ ( ( α = δ + O ( = α δ + O. + ( α δ + O (φ λ (x dx + δ ρ (R (φ λ ( + δ ( ρ O Cosequetly, Φ H = α δ + O ( = α + O δ (, which proves the formula (4.9. Now let β adα =. I this case from (3.9 we get (4.6 Φ H = (φ λ (x dx + δ δ { ( = α δ π( / + O { ( + δ α δ π( / + O ( = 4α δ 3. (π( / + O (φ λ (x dx + δ ρ (R (φ λ ( } 3 ( } ( 3 + O 4 From this it follows that Φ H = α ( δ π( / + O, which proves the formula (4.. The proofs for the other cases are similar. Theorem 4.3. The first compoets of the ormalized eigeelemets (4.4 have the followig asymptotic represetatio as : Case : If β ad α,the
14 79 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov (4.7 sg(α ( δ cos ψ (x = ( α sg cos δ δ π( (x + Case : If β ad α =,the (4.8 ψ (x sg( α ( δ si π = ( α sg si δ δ ( π( (x + ( (x + ( π( (x + Case 3: If β =ad α,the ( + O for x [,, ( + O for x (, ], ( + O for x [,, ( + O for x (, ], (4.9 δ f(xg(xdx + δ f(xg(xdx + δ ρ R (fg =, Case 4: If β =ad α,the (4. sg( α ( δ si ψ (x = ( sg α si δ δ π(x + ( π(x + ( + O ( + O for x [,, for x (, ]. Each of this asymptotic equalities hold uiformly for x. (Here, as usual, sg deotes the sig fuctio (4. Proof. Let β adα. I this case, from (4.9 it follows that ( δ = Φ H α + O. Puttig (3.8 ad (4. ito (4.3 we fid the required asymptotic formula (4.7. Similarly, we ca derive the other formulas (4.8 ( Gree fuctio, resolvet operator ad self-adjoitess of the problem Let A : H H be defied by (. ad (., ad let λ ot be a eigevalue of A. For fidig the resolvet operator R(λ, A =(λi A cosider the operator equatio (5. (λi AU = F
15 O discotiuous Sturm-Liouville problems with trasmissio coditios 793 for F = ( f(x f H. This operator equatio is equivalet to the ihomogeeous differetial equatio (5. u +(λ q(xu = f(x forx [, (, ] subject to ihomogeeous boudary coditios (5.3 (5.4 α u( + α u ( =, λ(β u( β u ( + (β u( β u ( = f ad homogeeous trasmissio coditios (5.5 (5.6 u( δu(+ =, u ( δu (+ =. By applyig the same techiques as i our previous paper [4] we ca prove that the problem (5. (5.6 has a uique solutio u(x, λ, which ca be represeted as (5.7 χ (x, λ ω (λ u(x, λ = χ (x, λ ω (λ x φ (y, λf(ydy+ φ (x, λ ω (λ + δ δ φ (y, λf(ydy+ + φ (x, λ ω (λ x χ (y, λf(ydy χ (y, λf(ydy + δ f for x [,, x x φ (y, λf(ydy χ (y, λf(ydy+f for x (, ]. Deotig χ(x, λφ(y, λ δ for y x, ω(λ (5.8 G(x, y, λ = φ(x, λχ(y, λ δ for x y, ω(λ where x ady, the formula (5.7 reduced to (5.9 u(x, λ = δ G(x, y, λf(ydy + δ G(x, y, λf(ydy + δ f φ(x, λ ω(λ. O the other had, by applyig the fuctioal R to the Gree fuctio with respect to the variable y ad recallig that χ(x, λ satisfies the iitial
16 794 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov coditios (3.5 ad (3.6 we have (5. R (G(x, ; λ = β G(x, ; λ β G(x, ; λ y φ(x, λ = δ ω(λ (β χ(,λ β χ (,λ φ(x, λ = δ ω(λ (β (β λ + β β (β λ + β φ(x, λ = δ ρ ω(λ. Substitutig this ito (5.9 gives (5. u(x, λ = δ G(x, y, λf(ydy + δ + δ ρ R (G(x,,λf. G(x, y, λf(ydy Now itroducig ( G(x,,λ (5. G x,λ = R (G(x,,λ which we call the Gree elemet of the problem (5. (5.6, the last formula (5. takes the form (5.3 u(x, λ = G x,λ, F, where by F we mea ( f(x F =. Now we ca fid the resolvet operator of A i terms of Gree elemet G x,λ. As the fuctio u(x, λ defied by (5. is the solutio of the ihomogeeous boudary-trasmissio problem (5. (5.6 which is equivalet to the operator equatio (5. we have (5.4 R(λ, AF = for arbitrary F H. f ( ( u(x, λ Gx,λ, F R (u(,λ = R G,λ, F Theorem 5.. The operator A is self-adjoit o the Hilbert space H.
17 O discotiuous Sturm-Liouville problems with trasmissio coditios 795 Proof. First, we prove that A is desely defied o H. For this suppose ( g(x G = H g is orthogoal to D(A, i.e. (5.5 δ f(xg(xdx + δ f(xg(xdx + δ ρ R (fg = for all F ( f(x R D(A. Let C (f ([, (, ] be a set of ifiitely differetiable fuctios o [, (, ], each elemet of which vaishes o some eighborhood of the poits x =, x =adx =. It is clear from the defiitio of D(A thatc ([, (, ] {} D(A. By writig (5.5 for all F C ([, (, ] we ca see that g(x is orthogoal to C ([, (, ] i L (, with respect to the followig ier product δ f(xg(xdx + δ f(xg(xdx =forallf C ([, (, ]. Cosequetly, g(x vaishes, sice L (, is complete with respect to the above ier product. The, substitutig g(x = ito (5.5 yields (5.6 R (fg =, for all f L (, such that ( f(x R (f D(A. Choosig F = ( f (x R (f D(A such that R (f =, we have from (5.6 that g =. Cosequetly,G =, so D(A isdeseih. Further, sice A is symmetric it is eough to prove that D(A =D(A, where A is adjoit of A. LetF D(A. We must show that F D(A. By defiitio of A (5.7 AG, F H = G, A F H for all G D(A. From this it follows that (5.8 (ii AG, F = G, ( ii A F. We already kow that (see (5.4 λ = i is regular poit of A ad therefore we ca let (5.9 U = R( i, A( if A F,
18 796 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov that is (5. ( ii AU = if A F. Substitutig this ito (5.8 ad takig ito accout that A is symmetric ad U D(A wehave Cosequetly, (ii AG, F H = G, ( ii AU H = G, iu H G, AU H = ig, U H AG, U H = (ii AG, U H. (5. (ii AG, F U H =forallg H. Sice λ = i is regular poit of A we ca choose G = R(i, A(F U. Substitutig this ito (5. we get F U H =, so F = U ad therefore F D(A. The proof is complete. Departmet of Mathematics Faculty of Arts ad Scieces Gaziosmapasa Uiversity Tokat, Turkey omukhtarov@yahoo.com Departmet of Mathematics Faculty of Arts ad Scieces Odokuz Mayıs Uiversity 5539 Kurupelit-Samsu, Turkey mkadakal@yahoo.com Mathematic ad Mechaic Istitute Azerbaija Sciece Academy Baku, Azerbaija muhtarov@gop.edu.tr Refereces [] G.D.Birkhoff,O the asymptotic character of the solutio of the certai liear differetial equatios cotaiig parameter, Tras. Amer. Soc. 9 (98, 9 3.
19 O discotiuous Sturm-Liouville problems with trasmissio coditios 797 [] C. T. Fulto, Two-poit boudary value problems with eigevalue parameter cotaied i the boudary coditios, Proc. Roy. Soc. Ediburgh 77A (977, [3] D.B.Hito,A expasio theorem for a eigevalue problem with eigevalue parameter i the boudary coditio, Quart.J.Math.Oxford3 (979, [4] M. Kadakal, F. S. Muhtarov ad O. Sh. Mukhtarov, Greefuctioofoe discotiuous boudary value problem with trasmissio coditios, Bull. Pure Appl. Sci. E ( (, [5] O. Sh. Mukhtarov ad H. Demir, Coerciveess of the discotiuous iitialboudary value problem for parabolic equatios, IsraelJ.Math.4 (999, [6] O. Sh. Mukhtarov, M. Kademir ad N. Kuruoglu, Distributio of eigevalues for the discotiuous boudary value problem with fuctioal maypoit coditios, IsraelJ.Math.9 (, [7] O. Sh. Mukhtarov ad S. Yakubov, Problems for ordiary differetial equatios with trasmissio coditios, Appl. Aal. 8 (, [8] A. Scheider, A ote o eigevalue problems with eigevalue parameter i the boudary coditios, Math.Z.36 (974, [9] A. A. Shkalikov, Boudary value problems for ordiary differetial equatios with a parameter i boudary coditio, Trudy Sem. Imey I. G. Petrowsgo 9 (983, 9 9. [] A. N. Tikhoov ad A. A. Samarskii, Equatios of Mathematical Physics, Oxford ad New York, Pergamo, 963. [] E. C. Titchmarsh, Eigefuctios Expasio Associated With Secod Order Differetial Equatios I, d ed, Oxford Uiv. Press, Lodo, 96. [] I. Titeux ad S. Yakubov, Applicatio of Abstract Differetial Equatios to some Mechaical Problems, Kluwer Academic Publishers, Dordrecht, Bosto, Lodo, 3. [3] E. Tuc ad O. Sh. Mukhtarov, Fudametal solutios ad eigevalues of oe boudary-value problem with trasmissio coditios, Applied Mathematics ad Computatio, 3. [4] J. Walter, Regular eigevalue problems with eigevalue parameter i the boudary coditios, Math.Z.33 (973, 3 3. [5] S. Yakubov, Completeess of Root Fuctios of Regular Differetial Operators, Logma, Scietific Techical, New York, 994.
20 798 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov [6] S. Yakubov ad Y. Yakubov, Abel basis of root fuctios of regular boudary value problems, Math.Nachr.97 (999, [7], Differetial-Operator Equatios, Ordiary ad Partial Differetial Equatios, Chapma ad Hall/CRC, Boca Rato,, p. 568.
Inverse Nodal Problems for Differential Equation on the Half-line
Australia Joural of Basic ad Applied Scieces, 3(4): 4498-4502, 2009 ISSN 1991-8178 Iverse Nodal Problems for Differetial Equatio o the Half-lie 1 2 3 A. Dabbaghia, A. Nematy ad Sh. Akbarpoor 1 Islamic
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationMath 220B Final Exam Solutions March 18, 2002
Math 0B Fial Exam Solutios March 18, 00 1. (1 poits) (a) (6 poits) Fid the Gree s fuctio for the tilted half-plae {(x 1, x ) R : x 1 + x > 0}. For x (x 1, x ), y (y 1, y ), express your Gree s fuctio G(x,
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationSPECTRUM OF THE DIRECT SUM OF OPERATORS
Electroic Joural of Differetial Equatios, Vol. 202 (202), No. 20, pp. 8. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.ut.edu ftp ejde.math.txstate.edu SPECTRUM OF THE DIRECT SUM
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationExistence of viscosity solutions with asymptotic behavior of exterior problems for Hessian equations
Available olie at www.tjsa.com J. Noliear Sci. Appl. 9 (2016, 342 349 Research Article Existece of viscosity solutios with asymptotic behavior of exterior problems for Hessia equatios Xiayu Meg, Yogqiag
More informationNumerical Method for Blasius Equation on an infinite Interval
Numerical Method for Blasius Equatio o a ifiite Iterval Alexader I. Zadori Omsk departmet of Sobolev Mathematics Istitute of Siberia Brach of Russia Academy of Scieces, Russia zadori@iitam.omsk.et.ru 1
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationDETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS.
ICSV4 Cairs Australia 9- July 7 DTRMINATION OF MCHANICAL PROPRTIS OF A NON- UNIFORM BAM USING TH MASURMNT OF TH XCITD LONGITUDINAL LASTIC VIBRATIONS Pavel Aokhi ad Vladimir Gordo Departmet of the mathematics
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationPOWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS
Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationMETHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS
Please cite this article as: Staisław Kula, Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais, Scietific Research of the Istitute of Mathematics ad Computer Sciece, 009,
More informationModified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations
Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-
More informationWhere do eigenvalues/eigenvectors/eigenfunctions come from, and why are they important anyway?
Where do eigevalues/eigevectors/eigeuctios come rom, ad why are they importat ayway? I. Bacgroud (rom Ordiary Dieretial Equatios} Cosider the simplest example o a harmoic oscillator (thi o a vibratig strig)
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationSubject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points
Power series solutio of Differetial equatios about ordiary poits Subject: Differetial Equatios & Mathematical Modelig -III Lesso: Power series solutios of Differetial Equatios about ordiary poits Lesso
More informationPoincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains
Advaces i Pure Mathematics 23 3 72-77 http://dxdoiorg/4236/apm233a24 Published Olie Jauary 23 (http://wwwscirporg/oural/apm) Poicaré Problem for Noliear Elliptic Equatios of Secod Order i Ubouded Domais
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More informationNotes 18 Green s Functions
ECE 638 Fall 017 David R. Jackso Notes 18 Gree s Fuctios Notes are from D. R. Wilto, Dept. of ECE 1 Gree s Fuctios The Gree's fuctio method is a powerful ad systematic method for determiig a solutio to
More informationf(w) w z =R z a 0 a n a nz n Liouville s theorem, we see that Q is constant, which implies that P is constant, which is a contradiction.
Theorem 3.6.4. [Liouville s Theorem] Every bouded etire fuctio is costat. Proof. Let f be a etire fuctio. Suppose that there is M R such that M for ay z C. The for ay z C ad R > 0 f (z) f(w) 2πi (w z)
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationQuestion 1: The magnetic case
September 6, 018 Corell Uiversity, Departmet of Physics PHYS 337, Advace E&M, HW # 4, due: 9/19/018, 11:15 AM Questio 1: The magetic case I class, we skipped over some details, so here you are asked to
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationMATH4822E FOURIER ANALYSIS AND ITS APPLICATIONS
MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if
More informationTaylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationReconstruction of the Volterra-type integro-differential operator from nodal points
Keski Boudary Value Problems 18 18:47 https://doi.org/1.1186/s13661-18-968- R E S E A R C H Ope Access Recostructio of the Volterra-type itegro-differetial operator from odal poits Baki Keski * * Correspodece:
More informationSimilarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationChapter 3 Inner Product Spaces. Hilbert Spaces
Chapter 3 Ier Product Spaces. Hilbert Spaces 3. Ier Product Spaces. Hilbert Spaces 3.- Defiitio. A ier product space is a vector space X with a ier product defied o X. A Hilbert space is a complete ier
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
More informationA NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim
Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationA) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.
M.A./M.Sc. (Mathematics) Etrace Examiatio 016-17 Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More information8. Applications To Linear Differential Equations
8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationLinear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy
Liear Differetial Equatios of Higher Order Basic Theory: Iitial-Value Problems d y d y dy Solve: a( ) + a ( )... a ( ) a0( ) y g( ) + + + = d d d ( ) Subject to: y( 0) = y0, y ( 0) = y,..., y ( 0) = y
More informationarxiv: v1 [math.dg] 27 Jul 2012
ESTIMATES FOR EIGENVALUES OF THE PANEITZ OPERATOR* arxiv:107.650v1 [math.dg] 7 Jul 01 QING-MING CHENG Abstract. For a -dimesioal compact submaifold i the Euclidea space R N, we study estimates for eigevalues
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationEXTRINSIC ESTIMATES FOR EIGENVALUES OF THE LAPLACE OPERATOR. 1. introduction
EXTRINSIC ESTIMATES FOR EIGENVALUES OF THE LAPLACE OPERATOR DAGUANG CHEN AND QING-MING CHENG* Abstract. For a bouded domai i a complete Riemaia maifold M isometrically immersed i a Euclidea space, we derive
More informationf(x)g(x) dx is an inner product on D.
Ark9: Exercises for MAT2400 Fourier series The exercises o this sheet cover the sectios 4.9 to 4.13. They are iteded for the groups o Thursday, April 12 ad Friday, March 30 ad April 13. NB: No group o
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationBESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES
Proceedigs of the Ediburgh Mathematical Society 007 50, 3 36 c DOI:0.07/S00309505000 Prited i the Uited Kigdom BESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES SENKA BANIĆ, DIJANA ILIŠEVIĆ
More informationBangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010)
O Cesáro Meas of Order μ for Outer Fuctios ISSN 1749-3889 (prit), 1749-3897 (olie) Iteratioal Joural of Noliear Sciece Vol9(2010) No4,pp455-460 Maslia Darus 1, Rabha W Ibrahim 2 1,2 School of Mathematical
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Stirlig ad Lagrage Sprig 2003 This sectio of the otes cotais proofs of Stirlig s formula ad the Lagrage Iversio Formula. Stirlig s formula Theorem 1 (Stirlig s
More informationON BASICITY OF EIGENFUNCTIONS OF SECOND ORDER DISCONTINUOUS DIFFERENTIAL OPERATOR
ISSN 2304-0122 Ufa Mathematical Joural Vol 9 No 1 (2017) P 109-122 doi:1013108/2017-9-1-109 UDC 5179842 ON BASICITY OF EIGENFUNCTIONS OF SECOND ORDER DISCONTINUOUS DIFFERENTIAL OPERATOR BT BILALOV, TB
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationPAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu
More informationChapter 10 Partial Differential Equations and Fourier Series
Math-33 Chapter Partial Differetial Equatios November 6, 7 Chapter Partial Differetial Equatios ad Fourier Series Math-33 Chapter Partial Differetial Equatios November 6, 7. Boudary Value Problems for
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationNumerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets
Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More informationON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS * M. JA]IMOVI], I. KRNI] 1.
Yugoslav Joural of Operatios Research 1 (00), Number 1, 49-60 ON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS M. JA]IMOVI], I. KRNI] Departmet of Mathematics
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationSome Tauberian theorems for weighted means of bounded double sequences
A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. Some Tauberia theorems for weighted meas of bouded double sequeces Cemal Bele Received: 22.XII.202 / Revised: 24.VII.203/ Accepted: 3.VII.203
More informationON THE EXISTENCE OF E 0 -SEMIGROUPS
O HE EXISECE OF E -SEMIGROUPS WILLIAM ARVESO Abstract. Product systems are the classifyig structures for semigroups of edomorphisms of B(H), i that two E -semigroups are cocycle cojugate iff their product
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationA Negative Result. We consider the resolvent problem for the scalar Oseen equation
O Osee Resolvet Estimates: A Negative Result Paul Deurig Werer Varhor 2 Uiversité Lille 2 Uiversität Kassel Laboratoire de Mathématiques BP 699, 62228 Calais cédex Frace paul.deurig@lmpa.uiv-littoral.fr
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
More informationImplicit function theorem
Jovo Jaric Implicit fuctio theorem The reader kows that the equatio of a curve i the x - plae ca be expressed F x, =., this does ot ecessaril represet a fuctio. Take, for example F x, = 2x x =. (1 either
More informationGamma Distribution and Gamma Approximation
Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract
More informationLocal Approximation Properties for certain King type Operators
Filomat 27:1 (2013, 173 181 DOI 102298/FIL1301173O Published by Faculty of Scieces ad athematics, Uiversity of Niš, Serbia Available at: http://wwwpmfiacrs/filomat Local Approimatio Properties for certai
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationOn the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method ABSTRACT 1.
Malaysia Joural of Mathematical Scieces 9(): 337-348 (05) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Joural homepage: http://eispemupmedumy/joural O the Weak Localizatio Priciple of the Eigefuctio Expasios
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationExact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method
Exact Solutios for a Class of Noliear Sigular Two-Poit Boudary Value Problems: The Decompositio Method Abd Elhalim Ebaid Departmet of Mathematics, Faculty of Sciece, Tabuk Uiversity, P O Box 741, Tabuki
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationA note on the p-adic gamma function and q-changhee polynomials
Available olie at wwwisr-publicatioscom/jmcs J Math Computer Sci, 18 (2018, 11 17 Research Article Joural Homepage: wwwtjmcscom - wwwisr-publicatioscom/jmcs A ote o the p-adic gamma fuctio ad q-chaghee
More informationSome families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions
J. Math. Aal. Appl. 297 2004 186 193 www.elsevier.com/locate/jmaa Some families of geeratig fuctios for the multiple orthogoal polyomials associated with modified Bessel K-fuctios M.A. Özarsla, A. Altı
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationA NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p
A NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p SEICK KIM Abstract. Assume that Ω is a bouded domai i R with 2. We study positive solutios to the problem, u = u p i Ω, u(x) as x Ω, where p > 1. Such solutios
More informationPeriod Function of a Lienard Equation
Joural of Mathematical Scieces (4) -5 Betty Joes & Sisters Publishig Period Fuctio of a Lieard Equatio Khalil T Al-Dosary Departmet of Mathematics, Uiversity of Sharjah, Sharjah 77, Uited Arab Emirates
More informationFourier Series and the Wave Equation
Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig
More information